(4K_1,P_4)--free graphs

Speed

The speed of a class $X$ is the function $n \mapsto |X_n|,ドル where $X_n$ is the set of $n$-vertex labeled graphs in $X$.

Depending on the rate of growths of the speed of the class, ISGCI distinguishes the following values of the parameter:
Constant
Polynomial
Exponential
Factorial
Superfactorial (2ドル^{o(n^2)}$ )
Superfactorial (2ドル^{\Theta(n^2)}$ )

factorial

at most factorial From bounded cliquewidth, maximum degree or degeneracy.
at least factorial on (3K₁,P₃)-free

[1785]

V.E. Alekseev
On lower layers of a lattice of hereditary classes of graphs
Diskretn. Anal. Issled. Oper. Ser. 1 4:1 3-12 (1997)

[1786]

E.R. Scheinerman, J. Zito
On the size of hereditary classes of graphs
J. Combin. Th. B Vol. 61 No.1 16-39 (1994)

[1787]

J. Balogh, B. Bollobas, D. Weinreich
The speed of hereditary properties of graphs
J. Combin. Th. B Vol. 79 No.2 131-156 (2000)

acyclic chromatic number

[?]

The acyclic chromatic number of a graph $G$ is the smallest size of a vertex partition $\{V_1,\dots,V_l\}$ such that each $V_i$ is an independent set and for all $i,j$ that graph $G[V_i\cup V_j]$ does not contain a cycle.

Unbounded

Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

bandwidth

[?]

The bandwidth of a graph $G$ is the shortest maximum "length" of an edge over all one dimensional layouts of $G$. Formally, bandwidth is defined as $\min_{i \colon V \rightarrow \mathbb{N}\;}\{\max_{\{u,v\}\in E\;} \{|i(u)-i(v)|\}\mid i\text{ is injective}\}$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from cutwidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from treewidth

Unbounded on complete [by definition]

book thickness

[?]

A book embedding of a graph $G$ is an embedding of $G$ on a collection of half-planes (called pages) having the same line (called spine) as their boundary, such that the vertices all lie on the spine and there are no crossing edges. The book thickness of a graph $G$ is the smallest number of pages over all book embeddings of $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

Unbounded on complete

[1778]

F. Bernhart, P.C. Kainen
The book thickness of a graph
J. of Combin. Th. (B) 27 320-331 (1979)

booleanwidth

[?]

Consider the following decomposition of a graph $G$ which is defined as a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. The function $\text{cut-bool} \colon 2^{V(G)} \rightarrow R$ is defined as $\text{cut-bool}(A)$ := $\log_2|\{S \subseteq V(G) \backslash A \mid \exists X \subseteq A \colon S = (V(G) \backslash A) \cap \bigcup_{x \in X} N(x)\}|$. Every edge $e$ in $T$ partitions the vertices $V(G)$ into $\{A_e,\overline{A_e}\}$ according to the leaves of the two connected components of $T - e$. The booleanwidth of the above decomposition $(T,L)$ is $\max_{e \in E(T)\;} \{ \text{cut-bool}(A_e)\}$. The booleanwidth of a graph $G$ is the minimum booleanwidth of a decomposition of $G$ as above.

Bounded

Bounded from booleanwidth on the complement
Bounded from cliquewidth
Bounded from rankwidth

branchwidth

[?]

A branch decomposition of a graph $G$ is a pair $(T,\chi),ドル where $T$ is a binary tree and $\chi$ is a bijection, mapping leaves of $T$ to edges of $G$. Any edge $\{u, v\}$ of the tree divides the tree into two components and divides the set of edges of $G$ into two parts $X, E \backslash X,ドル consisting of edges mapped to the leaves of each component. The width of the edge $\{u,v\}$ is the number of vertices of $G$ that is incident both with an edge in $X$ and with an edge in $E \backslash X$. The width of the decomposition $(T,\chi)$ is the maximum width of its edges. The branchwidth of the graph $G$ is the minimum width over all branch-decompositions of $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from treewidth

carvingwidth

[?]

Consider a decomposition $(T,\chi)$ of a graph $G$ where $T$ is a binary tree with $|V(G)|$ leaves and $\chi$ is a bijection mapping the leaves of $T$ to the vertices of $G$. Every edge $e \in E(T)$ of the tree $T$ partitions the vertices of the graph $G$ into two parts $V_e$ and $V \backslash V_e$ according to the leaves of the two connected components in $T - e$. The width of an edge $e$ of the tree is the number of edges of a graph $G$ that have exactly one endpoint in $V_e$ and another endpoint in $V \backslash V_e$. The width of the decomposition $(T,\chi)$ is the largest width over all edges of the tree $T$. The carvingwidth of a graph is the minimum width over all decompositions as above.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from treewidth

chromatic number

[?]

The chromatic number of a graph is the minimum number of colours needed to label all its vertices in such a way that no two vertices with the same color are adjacent.

Unbounded

Unbounded from maximum clique

cliquewidth

[?]

The cliquewidth of a graph is the number of different labels that is needed to construct the graph using the following operations:

creation of a vertex with label $i,ドル
disjoint union,
renaming labels $i$ to label $j,ドル and
connecting all vertices with label $i$ to all vertices with label $j$.

Bounded

Bounded from booleanwidth
Bounded from cliquewidth on the complement
Bounded from rankwidth

Bounded on (5,1)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Bounded on (6,2)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Bounded on (7,3)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Bounded on (9,6)

[488]

A. Gy\'arf\'as, D. Kratsch, J. Lehel, F. Maffray
Minimal non--neighbourhood--perfect graphs
J. Graph Theory 21 55--66 1996

Bounded on (P,P,co-fork,fork)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Bounded on P₄-tidy

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Bounded on (P₅,bull,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1187]

J.-L. Fouquet
A decomposition for a class of (P_5, \overline{P_5})-free graphs.
Discrete Math. 121 (1993) 19-30

Bounded on (P₅,fork,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[402]

J.L. Fouquet, V. Giakoumakis
On semi--$P_4$--sparse graphs
Discrete Math. 165/166 1997 277--300

Bounded on (P₅,gem)-free

[1171]

A. Brandstaedt, D. Kratsch
On the structure of (P_5,gem)-free graphs.
Manuscript 2002

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1189]

A. Brandstaedt, H.-O. Le, R. Mosca
Chordal co-gem-free graphs have bounded clique-width
Manuscript 2002

Bounded on (P,fork,gem)-free

[1184]

A. Brandstaedt, H.-O. Le, J.M. Vanherpe
Structure and stability number of (chair, co-P, gem)-free graphs.
Manuscript 2001

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Bounded on (P,fork,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1186]

A. Brandstaedt, R. Mosca
On variations of P_4-sparse graphs.
Manuscript 2001

Bounded on (bull,co-fork,fork)-free

[1124]

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Bounded on (bull,fork,gem)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Bounded on (bull,fork,house)-free

[1124]

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Bounded on cliquewidth 2
Bounded on cliquewidth 3
Bounded on cliquewidth 4
Bounded on (co-gem,gem)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1188]

A. Brandstaedt, H.-O. Le, R. Mosca
(gem, co-gem)-free graphs have bounded clique-width
Manuscript 2002

Bounded on distance-hereditary

[1177]

Golumbic, Martin Charles; Rotics, Udi
On the clique-width of perfect graph classes (extended abstract) .
Graph theoretic concepts in computer science. 25th international workshop, WG '99 Ascona, Switzerland, June 17-19, 1999. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1665, 135-147 (1999)

Bounded on partner-limited

[1179]

J.M. Vanherpe
Clique-width of partner-limited graphs.
International Conference of Graph Theory, Marseille 2000

Bounded on probe P₄-reducible
Bounded on probe P₄-sparse
Bounded on probe distance-hereditary
Bounded on (q, q-3), fixed q>= 7

[1176]

J.A. Makowsky, U. Rotics
On the clique-width of graphs with few $P_4$'s.
International Journal of Foundations of Computer Science 10 (1999) 329-348

Bounded on (q,q-4), fixed q

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Bounded on semi-P₄-sparse

[1186]

A. Brandstaedt, R. Mosca
On variations of P_4-sparse graphs.
Manuscript 2001

Bounded on tree-cograph

cochromatic number

[?]

The cochromatic number of a graph $G$ is the minimum number of colours needed to label all its vertices in such a way that that every set of vertices with the same colour is either independent in G, or independent in $\overline{G}$.

Bounded

Bounded from minimum clique cover

Bounded on (p,q<=2)-colorable

[1866]

(no preview available)

cutwidth

[?]

The cutwidth of a graph $G$ is the smallest integer $k$ such that the vertices of $G$ can be arranged in a linear layout $v_1, \ldots, v_n$ in such a way that for every $i = 1, \ldots,n - 1,ドル there are at most $k$ edges with one endpoint in $\{v_1, \ldots, v_i\}$ and the other in $\{v_{i+1}, \ldots, v_n\}$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from treewidth

degeneracy

[?]

Let $G$ be a graph and consider the following algorithm:

Find a vertex $v$ with smallest degree.
Delete vertex $v$ and its incident edges.
Repeat as long as the graph is not empty.

The degeneracy of a graph $G$ is the maximum degree of a vertex when it is deleted in the above algorithm.

Unbounded

Unbounded from chromatic number
Unbounded from maximum clique

diameter

[?]

The diameter of a graph $G$ is the length of the longest shortest path between any two vertices in $G$.

Bounded

Bounded from distance to cograph
Bounded from maximum independent set
Bounded from maximum induced matching
Bounded from minimum clique cover
Bounded from minimum dominating set

Bounded on P₇-free [by definition]

distance to block

[?]

The distance to block of a graph $G$ is the size of a smallest vertex subset whose deletion makes $G$ a block graph.

Unbounded

Unbounded on (3K₁,P₃)-free

[1827]

(no preview available)

distance to clique

[?]

Let $G$ be a graph. Its distance to clique is the minimum number of vertices that have to be deleted from $G$ in order to obtain a clique.

Unbounded

Unbounded from distance to block
Unbounded from distance to cluster
Unbounded from distance to co-cluster

distance to cluster

[?]

A cluster is a disjoint union of cliques. The distance to cluster of a graph $G$ is the size of a smallest vertex subset whose deletion makes $G$ a cluster graph.

Unbounded

Unbounded from distance to block

distance to co-cluster

[?]

The distance to co-cluster of a graph is the minimum number of vertices that have to be deleted to obtain a co-cluster graph.

Unbounded

Unbounded from distance to cluster on the complement

distance to cograph

[?]

The distance to cograph of a graph $G$ is the minimum number of vertices that have to be deleted from $G$ in order to obtain a cograph .

Bounded

Bounded on cograph [by definition]

distance to linear forest

[?]

The distance to linear forest of a graph $G = (V, E)$ is the size of a smallest subset $S$ of vertices, such that $G[V \backslash S]$ is a disjoint union of paths and singleton vertices.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from pathwidth
Unbounded from treewidth

distance to outerplanar

[?]

The distance to outerplanar of a graph $G = (V,E)$ is the minumum size of a vertex subset $X \subseteq V,ドル such that $G[V \backslash X]$ is a outerplanar graph.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from treewidth

genus

[?]

The genus $g$ of a graph $G$ is the minimum number of handles over all surfaces on which $G$ can be embedded without edge crossings.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

max-leaf number

[?]

The max-leaf number of a graph $G$ is the maximum number of leaves in a spanning tree of $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from bandwidth
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from cutwidth
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from treewidth

maximum clique

[?]

The parameter maximum clique of a graph $G$ is the largest number of vertices in a complete subgraph of $G$.

Unbounded

Unbounded on complete [by definition]

maximum degree

[?]

The maximum degree of a graph $G$ is the largest number of neighbors of a vertex in $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

Unbounded on complete [by definition]

maximum independent set

[?]

An independent set of a graph $G$ is a subset of pairwise non-adjacent vertices. The parameter maximum independent set of graph $G$ is the size of a largest independent set in $G$.

Bounded

Bounded from maximum clique on the complement
Bounded from minimum clique cover

Bounded on 4K₁-free [by definition]
Bounded on 5K₁-free [by definition]

maximum induced matching

[?]

For a graph $G = (V,E)$ an induced matching is an edge subset $M \subseteq E$ that satisfies the following two conditions: $M$ is a matching of the graph $G$ and there is no edge in $E \backslash M$ connecting any two vertices belonging to edges of the matching $M$. The parameter maximum induced matching of a graph $G$ is the largest size of an induced matching in $G$.

Bounded

Bounded from maximum independent set
Bounded from minimum clique cover

maximum matching

[?]

A matching in a graph is a subset of pairwise disjoint edges (any two edges that do not share an endpoint). The parameter maximum matching of a graph $G$ is the largest size of a matching in $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to cluster
Unbounded from distance to co-cluster
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from pathwidth
Unbounded from tree depth
Unbounded from treewidth
Unbounded from vertex cover

minimum clique cover

[?]

A clique cover of a graph $G = (V, E)$ is a partition $P$ of $V$ such that each part in $P$ induces a clique in $G$. The minimum clique cover of $G$ is the minimum number of parts in a clique cover of $G$. Note that the clique cover number of a graph is exactly the chromatic number of its complement.

Bounded

Bounded from chromatic number on the complement

minimum dominating set

[?]

A dominating set of a graph $G$ is a subset $D$ of its vertices, such that every vertex not in $D$ is adjacent to at least one member of $D$. The parameter minimum dominating set for graph $G$ is the minimum number of vertices in a dominating set for $G$.

Bounded

Bounded from maximum independent set
Bounded from minimum clique cover

pathwidth

[?]

A path decomposition of a graph $G$ is a pair $(P,X)$ where $P$ is a path with vertex set $\{1, \ldots, q\},ドル and $X = \{X_1,X_2, \ldots ,X_q\}$ is a family of vertex subsets of $V(G)$ such that:

$\bigcup_{p \in \{1,\ldots ,q\}} X_p = V(G)$
$\forall\{u,v\} \in E(G) \exists p \colon u, v \in X_p$
$\forall v \in V(G)$ the set of vertices $\{p \mid v \in X_p\}$ is a connected subpath of $P$.

The width of a path decomposition $(P,X)$ is max$\{|X_p| - 1 \mid p \in \{1,\ldots ,q\}\}$. The pathwidth of a graph $G$ is the minimum width among all possible path decompositions of $G$.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from treewidth

rankwidth

[?]

Let $M$ be the $|V| \times |V|$ adjacency matrix of a graph $G$. The cut rank of a set $A \subseteq V(G)$ is the rank of the submatrix of $M$ induced by the rows of $A$ and the columns of $V(G) \backslash A$. A rank decomposition of a graph $G$ is a pair $(T,L)$ where $T$ is a binary tree and $L$ is a bijection from $V(G)$ to the leaves of the tree $T$. Any edge $e$ in the tree $T$ splits $V(G)$ into two parts $A_e, B_e$ corresponding to the leaves of the two connected components of $T - e$. The width of an edge $e \in E(T)$ is the cutrank of $A_e$. The width of the rank-decomposition $(T,L)$ is the maximum width of an edge in $T$. The rankwidth of the graph $G$ is the minimum width of a rank-decomposition of $G$.

Bounded

Bounded from booleanwidth
Bounded from cliquewidth
Bounded from rankwidth on the complement

tree depth

[?]

A tree depth decomposition of a graph $G = (V,E)$ is a rooted tree $T$ with the same vertices $V,ドル such that, for every edge $\{u,v\} \in E,ドル either $u$ is an ancestor of $v$ or $v$ is an ancestor of $u$ in the tree $T$. The depth of $T$ is the maximum number of vertices on a path from the root to any leaf. The tree depth of a graph $G$ is the minimum depth among all tree depth decompositions.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from pathwidth
Unbounded from treewidth

treewidth

[?]

A tree decomposition of a graph $G$ is a pair $(T, X),ドル where $T = (I, F)$ is a tree, and $X = \{X_i \mid i \in I\}$ is a family of subsets of $V(G)$ such that

the union of all $X_i,ドル $i \in I$ equals $V,ドル
for all edges $\{v,w\} \in E,ドル there exists $i \in I,ドル such that $v, w \in X_i,ドル and
for all $v \in V$ the set of nodes $\{i \in I \mid v \in X_i\}$ forms a subtree of $T$.

The width of the tree decomposition is $\max |X_i| - 1$.
The treewidth of a graph is the minimum width over all possible tree decompositions of the graph.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

Unbounded on complete [by definition]

vertex cover

[?]

Let $G$ be a graph. Its vertex cover number is the minimum number of vertices that have to be deleted in order to obtain an independent set.

Unbounded

Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to cluster
Unbounded from distance to co-cluster
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from maximum matching
Unbounded from pathwidth
Unbounded from tree depth
Unbounded from treewidth

book thickness decomposition

[?]

Input: A graph G in this class and an integer k.

Output: True iff the book thickness of G is at most k.

Unknown to ISGCI

booleanwidth decomposition

[?]

Input: A graph G in this class.

Output: An expression that constructs G according to the rules for booleanwidth, using only a constant number of labels.
Undefined if this class has unbounded booleanwidth.

Polynomial

Polynomial from Bounded booleanwidth

Polynomial on circular permutation
Polynomial on circular trapezoid
Polynomial on d-trapezoid
Polynomial on permutation

cliquewidth decomposition

[?]

Input: A graph G in this class.

Output: An expression that constructs G according to the rules for cliquewidth, using only a constant number of labels.
Undefined if this class has unbounded cliquewidth.

Linear

Polynomial from Bounded cliquewidth
Polynomial from cliquewidth decomposition on the complement

Linear on (5,1)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Linear on (6,2)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Linear on (7,3)

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Linear on (9,6)

[488]

A. Gy\'arf\'as, D. Kratsch, J. Lehel, F. Maffray
Minimal non--neighbourhood--perfect graphs
J. Graph Theory 21 55--66 1996

Linear on (P,P,co-fork,fork)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Linear on P₄-tidy

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Linear on (P₅,bull,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1187]

J.-L. Fouquet
A decomposition for a class of (P_5, \overline{P_5})-free graphs.
Discrete Math. 121 (1993) 19-30

Linear on (P₅,fork,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[402]

J.L. Fouquet, V. Giakoumakis
On semi--$P_4$--sparse graphs
Discrete Math. 165/166 1997 277--300

Linear on (P,fork,gem)-free

[1184]

A. Brandstaedt, H.-O. Le, J.M. Vanherpe
Structure and stability number of (chair, co-P, gem)-free graphs.
Manuscript 2001

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Linear on (P,fork,house)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

[1186]

A. Brandstaedt, R. Mosca
On variations of P_4-sparse graphs.
Manuscript 2001

Linear on (bull,co-fork,fork)-free

[1124]

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Linear on (bull,fork,gem)-free

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Linear on (bull,fork,house)-free

[1124]

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

[1185]

A. Brandstaedt, F.F. Dragan, H.-O. Le, R. Mosca
New graph classes of bounded clique-width
WG 2002, Lecture Notes in Computer Science 2573, 57-67 (2002)

Linear on cliquewidth 2
Linear on distance-hereditary

[1177]

Golumbic, Martin Charles; Rotics, Udi
On the clique-width of perfect graph classes (extended abstract) .
Graph theoretic concepts in computer science. 25th international workshop, WG '99 Ascona, Switzerland, June 17-19, 1999. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1665, 135-147 (1999)

Linear on partner-limited

[1179]

J.M. Vanherpe
Clique-width of partner-limited graphs.
International Conference of Graph Theory, Marseille 2000

Linear on (q, q-3), fixed q>= 7

[1176]

J.A. Makowsky, U. Rotics
On the clique-width of graphs with few $P_4$'s.
International Journal of Foundations of Computer Science 10 (1999) 329-348

Linear on (q,q-4), fixed q

[1175]

B. Courcelle, J.A. Makowsky, U. Rotics
Linear time optimization problems on graphs of bounded clique width.
Theory of Computing Systems 33 (2000) 125-150

Linear on semi-P₄-sparse

[1186]

A. Brandstaedt, R. Mosca
On variations of P_4-sparse graphs.
Manuscript 2001

Polynomial [$O(V^2E)$] on cliquewidth 3

[1178]

Corneil, Derek G.; Habib, Michel; Lanlignel, Jean-Marc; Reed, Bruce; Rotics, Udi
Polynomial time recognition of clique-width $\leq 3$ graphs (extended abstract).
Gonnet, Gastón H. (ed.) et al., LATIN 2000: Theoretical informatics. 4th Latin American symposium, Punta del Este, Uruguay, April 10-14, 2000. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1776, 126-134 (2000)

Polynomial on tree-cograph

cutwidth decomposition

[?]

Input: A graph G in this class and an integer k.

Output: True iff the cutwidth of G is at most k.

Unknown to ISGCI

treewidth decomposition

[?]

Input: A graph G in this class and an integer k.

Output: True iff the treewidth of G is at most k.

Linear

Linear on distance-hereditary

[159]

H. Broersma, E. Dahlhaus, T. Kloks
A linear time algorithm for minimum fill--in and treewidth for distance--hereditary graphs
5th Twente workshop 1997

Linear on permutation

[1418]

D. Meister
Treewidth and minimum fill-in on permutation graphs in linear time
Theoretical Comp. Sci. 411 No. 40-42 3685-3700 (2010)

Polynomial on HHD-free

[1420]

H.J. Broersma, E. Dahlhaus, T. Kloks
Algorithms for the treewidth and minimum fill-in of HHD-free graphs
23rd Intern. Workshop on Graph--Theoretic Concepts in Comp. Sci. WG'97, Lecture Notes in Comp. Sci. 1335 (1997) 109-117

Polynomial on circle

[119]

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

Polynomial on circular permutation

[119]

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

Polynomial on co-comparability graphs of dimension d posets

[675]

T. Kloks, D. Kratsch, J. Spinrad
On treewidth and minimum fill--in of asteroidal triple--free graphs
Theor. Comp. Sci. 175 1997 309--335

Polynomial on cograph

[119]

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

Polynomial on d-trapezoid

[675]

T. Kloks, D. Kratsch, J. Spinrad
On treewidth and minimum fill--in of asteroidal triple--free graphs
Theor. Comp. Sci. 175 1997 309--335

Polynomial on d-trapezoid
Polynomial on permutation

[119]

H.L. Bodlaender
A tourist guide through treewidth
Acta Cybernetica 11 1993 1--23

Polynomial on (q,q-4), fixed q

[1422]

L. Babel
Triangulating graphs with few P4's
Disc. Appl. Math. 89 45-57 (1998)

Polynomial [$O(V^2)$] on trapezoid

[1417]

H.L. Bodlaender, T. Kloks, D. Kratsch, H. Mueller
Treewidth and minimum fill-in on d-trapezoid graphs
J. Graph Algorithms Appl. 2 1-23 (1998)

Polynomial [$O((V+E) \log V)$] on weak bipolarizable

[1419]

E. Dahlhaus
Minimum fill-in and treewidth on graphs modularly decomposable into chordal graphs
24th Intern. Workshop on Graph--Theoretic Concepts in Comp. Sci. WG'98, Lecture Notes in Comp. Sci. 1517 (1998) 351-358

Polynomial on weakly chordal

[1421]

V. Bouchitte, I. Todinca
Treewidth and minimum fill-in of weakly triangulated graphs
Annual symposium on theoretical aspects of computer science STACS 99, Lecture Notes in Comp. Sci. 1563 (1999) 197-206

3-Colourability

[?]

Input: A graph G in this class.

Output: True iff each vertex of G can be assigned one colour out of 3 such that whenever two vertices are adjacent, they have different colours.

Linear

Linear from Colourability
Linear from FPT-Linear on cliquewidth and Linear decomposition time
Linear from FPT-Linear on maximum independent set and Linear decomposition time
Polynomial from Colourability
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on minimum clique cover and Linear decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from FPT-Linear on cliquewidth and Polynomial decomposition time

Polynomial on 2P₃-free

[1431]

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

Polynomial on 4K₁-free

[1431]

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

Polynomial [$O(V^4)$] on AT-free

[1439]

J. Stacho
3-colouring of AT-free graphs in polynomial time
21st International Symposium on Algorithms and Computation ISAAC, Lecture Notes in Comp. Sci. LNCS 6507 144-155 (2010)

Polynomial on P₂ ∪ P₄-free

[1431]

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

Polynomial on P₅-free

[1242]

B. Randerath, I. Schiermeyer, M. Tewes
Three-colourability and forbidden subgraphs II: polynomial algorithms
Discrete Math. 251 137-212 (2002)

[1441]

C.T. Hoang, M. Kaminski, V. Lozin, J. Sawada, X. Shu
Deciding k-colorability of P_5-free graphs in polynomial time
Algorithmica Vol.57 No.1 74-81 (2010)

Polynomial on P₆-free

[1243]

B. Randerath, I. Schiermeyer
3-colorability in P for P_6-free graphs
Discrete Appl. Math. 136 299-313 (2004)

Polynomial on co-gem-free

[1431]

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

Polynomial on nP₃-free, fixed n

[1431]

H. Broersma, P.A. Golovach, D. Paulusma, J. Song
Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time
Manuscript (2010)

Polynomial on odd-hole-free

[1744]

(no preview available)

Clique

[?]

Input: A graph G in this class and an integer k.

Output: True iff G contains a set S of pairwise adjacent vertices, with |S| >= k.

Linear

Linear from Weighted clique
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on cliquewidth and Linear decomposition time
Polynomial from FPT on cliquewidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from Independent set on the complement
Polynomial from Weighted clique

Linear on Matula perfect

[221]

V. Chv\'atal, C.T. Ho\`ang, N.V.R. Mahadev, D. de Werra
Four classes of perfectly orderable graphs
J. Graph Theory 11 1987 481--495

Linear on Welsh-Powell perfect

[221]

V. Chv\'atal, C.T. Ho\`ang, N.V.R. Mahadev, D. de Werra
Four classes of perfectly orderable graphs
J. Graph Theory 11 1987 481--495

Linear on galled-tree explainable

[1875]

M. Hellmuth, G.E. Scholz
Solving NP-hard problems on GaTEx graphs: Linear-time algorithms for perfect orderings, cliques, colorings, and independent sets
Theoretical Comp. Sci. 1037 115-157 (2025)

Polynomial on b-perfect

[1478]

C.T. Hoang, F. Maffray, M. Mechebbek
A characterization of b-perfect graphs
Manuscript, 2010

Polynomial on biclique separable

[1304]

Eschen, Elaine M.; Hoàng, Chính T.; Petrick, Mark D.T.; Sritharan, R
Disjoint clique cutsets in graphs without long holes
J. Graph Theory 48, No.4, 277-298 (2005)

Polynomial [$O(V log^2 V)$] on circle

[1466]

A. Tiskin
Fast distance multiplication of unit-Monge matrices
Proc. of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms SODA 1287-1296 (2010)

Polynomial on circular perfect

[1408]

A. Pecher, A.K. Wagler
Clique and chromatic number of circular-perfect graphs
Proceedings of ISCO 2010 - International Symposium on Combinatorial Optimization, Elec. Notes in Discrete Math 36 199-206 (2010)

Polynomial [$O(V \log V)$] on multitolerance
Polynomial [$O(V \log V)$] on tolerance

Clique cover

[?]

Input: A graph G in this class and an integer k.

Output: True iff the vertices of G can be partitioned into k sets S_i, such that whenever two vertices are in the same set S_i, they are adjacent.

Polynomial

Polynomial from Colourability on the complement
Polynomial from XP on booleanwidth and Polynomial decomposition time
Polynomial from XP on cliquewidth and Linear decomposition time
Polynomial from XP on cliquewidth and Polynomial decomposition time
Polynomial from XP on rankwidth and Linear decomposition time

Polynomial [$O(V log^{d-1} V)$] on d-trapezoid
Polynomial [$O(V log log V)$] on trapezoid

Colourability

[?]

Input: A graph G in this class and an integer k.

Output: True iff each vertex of G can be assigned one colour out of k such that whenever two vertices are adjacent, they have different colours.

Linear

Polynomial from Clique cover on the complement
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on cliquewidth and Linear decomposition time
Polynomial from FPT on cliquewidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time

Linear on Matula perfect

[221]

V. Chv\'atal, C.T. Ho\`ang, N.V.R. Mahadev, D. de Werra
Four classes of perfectly orderable graphs
J. Graph Theory 11 1987 481--495

Linear on Welsh-Powell perfect

[221]

V. Chv\'atal, C.T. Ho\`ang, N.V.R. Mahadev, D. de Werra
Four classes of perfectly orderable graphs
J. Graph Theory 11 1987 481--495

Linear on galled-tree explainable

[1874]

M. Hellmuth, G.E. Scholz
Resolving prime modules: The structure of pseudo-cographs and galled-tree explainable graphs
Disc. Appl. Math. 343 25-43 (2024)

Polynomial on P₄-free

[1433]

D. Kral, J. Kratochvil, Z. Tuza, G.J. Woeginger
Complexity of coloring graphs without forbidden induced subgraphs
Proceedings of the 27th International Workshop on Graph-Theoretic Concepts in Computer Science WG'01, LNCS 2204, 254-262 (2001)

Polynomial on b-perfect

[1478]

C.T. Hoang, F. Maffray, M. Mechebbek
A characterization of b-perfect graphs
Manuscript, 2010

Polynomial on biclique separable

[1304]

Eschen, Elaine M.; Hoàng, Chính T.; Petrick, Mark D.T.; Sritharan, R
Disjoint clique cutsets in graphs without long holes
J. Graph Theory 48, No.4, 277-298 (2005)

Polynomial on circular perfect

[1408]

A. Pecher, A.K. Wagler
Clique and chromatic number of circular-perfect graphs
Proceedings of ISCO 2010 - International Symposium on Combinatorial Optimization, Elec. Notes in Discrete Math 36 199-206 (2010)

Polynomial [$O(V^3)$] on co-comparability

[451]

M.C. Golumbic
The complexity of comparability graph recognition and coloring
Computing 18 1977 199--208

Polynomial [$O(V^2)$] on comparability

[1424]

C.T. Hoang
Efficient algorithms for minimum weighted colouring of some classes of perfect graphs
Discrete Appl. Math. 55 133-143 (1994)

Polynomial [$O(V \log V)$] on multitolerance

[1497]

G.B. Mertzios
An intersection model for multitolerance graphs: Efficient algorithms and hierarchy
Proc. of 21 annual ACM-SIAM symposium on Discrete algorithms SODA2011 1306-1317 (2011)

Polynomial on perfect

[476]

M. Gr\"otschel, L. Lov\'asz, A. Schrijver
The ellipsoid method and its consequences in combinatorial optimization
Combinatorica 1 169--197, 1981 Corrigendum

Polynomial [$O(VE)$] on perfectly orderable
Polynomial [$O(V \log V)$] on permutation

[453]

M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980

Polynomial on permutation

[1165]

A. Brandstaedt, D. Kratsch
On the restriction of some NP-complete graph problems to permutation graphs.
Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 53-62 (1985)

Polynomial [$O(V \log V)$] on tolerance
Polynomial [$O(V log log V)$] on trapezoid
Polynomial [$O(V^4E)$] on weakly chordal

[530]

R.B. Hayward, C. Ho\`ang, F. Maffray
Optimizing weakly triangulated graphs
Graphs and Combinatorics 5 339--349, Erratum: 6 (1990) 33--35 1989

Domination

[?]

Input: A graph G in this class and an integer k.

Output: True iff G contains a set S of vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.

Linear

Linear from FPT-Linear on cliquewidth and Linear decomposition time
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from FPT-Linear on cliquewidth and Polynomial decomposition time
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time
Polynomial from XP on minimum dominating set and Linear decomposition time

Linear on distance-hereditary

[1153]

F. Nicolai, T. Szymczak
Homogeneous sets and domination: A linear time algorithm for distance-hereditary graphs
Networks 37,No.3 117-128 (2001)

Linear [$O(V)$] on permutation

[1147]

M. Farber, J.M. Keil
Domination in permutation graphs
J. Algorithms 6 (1985) 309-321

[1148]

K. Tsai, W.L. Hsu
Fast algorithms for the dominating set problem on permutation graphs
Algorithmica 9 (1993) 601-614

[1149]

C. Rhee, Y.D. Liang, S.K. Dhall, S. Lakshmivarahan
An O(n+m) algorithm for finding a minimum-weight dominating set in a permutation graph
SIAM J. Comput. 25 (1996) 404-419

[1165]

A. Brandstaedt, D. Kratsch
On the restriction of some NP-complete graph problems to permutation graphs.
Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 53-62 (1985)

[1342]

H.S. Chao, F.R. Hsu, R.C.T. Lee
An Optimal Algorithm for Finding the Minimum Cardinality Dominating Set on Permutation Graphs
Discrete Appl. Math. 102, No.3 159-173 (2000)

Polynomial on AT-free

[1152]

D. Kratsch
Domination and total domination in asteroidal triple-free graphs
Discrete Appl. Math. 99 No.1-3, 111-123 (2000)

Polynomial [$O(n^2 log^5 n)$] on co-bounded tolerance

[1172]

J. Mark Keil, P. Belleville
Dominating the complements of bounded tolerance graphs and the complements of trapezoid graphs.
Discrete Appl. Math. 140, No.1-3, 73-89 (2004). [ISSN 0166-218X]

Polynomial on co-comparability

[1150]

D. Kratsch, L. Stewart
Domination on cocomparability graphs
SIAM J. Discrete Math. 6(3) (1993) 400-417

[1151]

H. Breu, D.G. Kirkpatrick
Algorithms for the dominating set and Steiner set problems in cocomparability graphs
Manuscript 1993

Polynomial on cograph

[524]

T.W. Haynes, S.T. Hedetniemi, P.J. Slater (eds.)
Fundamentals of Domination in Graphs
Marcel Dekker, New York, Basel , Vol. 208 1998

Polynomial on k-polygon

[352]

E.S. Elmallah, L.K. Stewart
Independence and domination in polygon graphs
Discrete Appl. Math. 44 1993 65--77

Polynomial on permutation

[1165]

A. Brandstaedt, D. Kratsch
On the restriction of some NP-complete graph problems to permutation graphs.
Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 53-62 (1985)

Polynomial on trapezoid

[1155]

Liang, Y. Daniel
Dominations in trapezoid graphs.
Inf. Process. Lett. 52, No.6, 309-315 (1994)

Feedback vertex set

[?]

Input: A graph G in this class and an integer k.

Output: True iff G contains a set S of vertices, with |S| <= k, such that every cycle in G contains a vertex from S.

Linear

Linear from Weighted feedback vertex set
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on cliquewidth and Linear decomposition time
Polynomial from FPT on cliquewidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from Weighted feedback vertex set
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Polynomial [$O(V^8E^2)$] on AT-free

[1581]

Feedback vertex set on AT-free graphs
D. Kratsch, H. Mueller, I. Todinca
Discrete Appl. Math. 156 No. 10 1936-1947 (2008)

Polynomial [$O(V^2E)$] on co-comparability

[1579]

M.S Chang, Y.D. Liang
Minimum feedback vertex set in cocomparability graphs and convex bipartite graphs
Acta Informatica 34 337-346 (1997)

Polynomial [$O(V^4)$] on co-comparability

[1578]

S.R. Coorg, C.P. Rangan
Feedback vertex set on cocomparability graphs
Networks 26 101-111 (1995)

Polynomial [$O(VE)$] on permutation

[1576]

Y.D. Liang
On the feedback vertex set problem in permutation graphs
Information Proc. Lett. 52 123-129 (1994)

Polynomial [$O(VE^2)$] on permutation

[1165]

A. Brandstaedt, D. Kratsch
On the restriction of some NP-complete graph problems to permutation graphs.
Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 53-62 (1985)

Polynomial [$O(V^6)$] on permutation

[1165]

A. Brandstaedt, D. Kratsch
On the restriction of some NP-complete graph problems to permutation graphs.
Fundamentals of computation theory, Proc. 5th Int. Conf., Cottbus/Ger. 1985, Lect. Notes Comput. Sci. 199, 53-62 (1985)

Polynomial [$O(VE)$] on trapezoid

[1576]

Y.D. Liang
On the feedback vertex set problem in permutation graphs
Information Proc. Lett. 52 123-129 (1994)

Graph isomorphism

[?]

Input: Graphs G and H in this class

Output: True iff G and H are isomorphic.

Linear

Polynomial from Graph isomorphism on the complement
Polynomial from XP on booleanwidth and Polynomial decomposition time
Polynomial from XP on cliquewidth and Linear decomposition time
Polynomial from XP on cliquewidth and Polynomial decomposition time
Polynomial from XP on rankwidth and Linear decomposition time

Linear on permutation

[234]

C.J. Colbourn
On testing isomorphism of permutation graphs
Networks 11 1981 13--21

Polynomial on cograph

[251]

D.G. Corneil, H. Lerchs, L. Stewart--Burlingham
Complement reducible graphs
Discrete Appl. Math. 3 1981 163--174

Hamiltonian cycle

[?]

Input: A graph G in this class.

Output: True iff G has a simple cycle that goes through every vertex of the graph.

Linear

Polynomial from FPT on maximum independent set and Linear decomposition time
Polynomial from FPT on minimum clique cover and Linear decomposition time
Polynomial from XP on booleanwidth and Polynomial decomposition time
Polynomial from XP on cliquewidth and Linear decomposition time
Polynomial from XP on cliquewidth and Polynomial decomposition time
Polynomial from XP on rankwidth and Linear decomposition time

Linear on cograph

[251]

D.G. Corneil, H. Lerchs, L. Stewart--Burlingham
Complement reducible graphs
Discrete Appl. Math. 3 1981 163--174

Linear on distance-hereditary

[1534]

R.W. Hung, M.S. Chang
Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs
Theoret. Comput. Sci. 341 411-440 (2005)

[1535]

S.Y. Hsieh, C.W. Ho, T.S. Hsu, M.T. Ko
The Hamiltonian problem on distance-hereditary graphs
Discrete Appl. Math. 153 508-524 (2006)

Polynomial on 4K₁-free

[1861]

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

Polynomial on 5K₁-free

[1861]

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

Polynomial on co-comparability

[1524]

J.S. Deogun, G. Steiner
Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs
SIAM J. Computing 23 Issue 3 520-552 (1994)

Hamiltonian path

[?]

Input: A graph G in this class.

Output: True iff G has a simple path that goes through every vertex of the graph.

Linear

Polynomial from FPT on maximum independent set and Linear decomposition time
Polynomial from FPT on minimum clique cover and Linear decomposition time
Polynomial from XP on booleanwidth and Polynomial decomposition time
Polynomial from XP on cliquewidth and Linear decomposition time
Polynomial from XP on cliquewidth and Polynomial decomposition time
Polynomial from XP on rankwidth and Linear decomposition time

Linear on cograph

[251]

D.G. Corneil, H. Lerchs, L. Stewart--Burlingham
Complement reducible graphs
Discrete Appl. Math. 3 1981 163--174

Linear on distance-hereditary

[1534]

R.W. Hung, M.S. Chang
Linear-time algorithms for the Hamiltonian problems on distance-hereditary graphs
Theoret. Comput. Sci. 341 411-440 (2005)

Polynomial on 4K₁-free

[1861]

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

Polynomial on 5K₁-free

[1861]

N. Jedličková, J. Kratochvil
On the structure of Hamiltonian graphs with small independence number
Combinatorial Algorithms. IWOCA 2024. LNCS 14764 180-192 (2024)

Polynomial on co-comparability

[1524]

J.S. Deogun, G. Steiner
Polynomial Algorithms for Hamiltonian Cycle in Cocomparability Graphs
SIAM J. Computing 23 Issue 3 520-552 (1994)

Independent dominating set

[?]

Input: A graph G in this class and an integer k.

Output: True iff G contains a set S of pairwise non-adjacent vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.

Linear

Linear from Weighted independent dominating set
Polynomial from Weighted independent dominating set

Independent set

[?]

Input: A graph G in this class and an integer k.

Output: True iff G contains a set S of pairwise non-adjacent vertices, such that |S| >= k.

Linear

Linear from Weighted independent set
Polynomial from Clique on the complement
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on cliquewidth and Linear decomposition time
Polynomial from FPT on cliquewidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from Weighted independent set
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Linear on P₄-tidy

[440]

V. Giakoumakis, F. Roussel, H. Thuillier
On $P_4$--tidy graphs
Discrete Math. and Theor. Comp. Sci. 1 1997 17--41

Linear on co-comparability

[1100]

R.M. McConnell, J.P. Spinrad
Modular decomposition and transitive orientation
Discrete Math. 201 (1999) 189-241

Linear on extended P₄-laden

[438]

V. Giakoumakis
$P_4$--laden graphs: a new class of brittle graphs
Inf. Proc. Letters 60 1996 29--36

Linear on galled-tree explainable

[1875]

M. Hellmuth, G.E. Scholz
Solving NP-hard problems on GaTEx graphs: Linear-time algorithms for perfect orderings, cliques, colorings, and independent sets
Theoretical Comp. Sci. 1037 115-157 (2025)

Linear on partner-limited

[1180]

F. Roussel, I. Rusu, H. Thuillier
On graphs with limited number of P_4-partners.
International Journal of Foundations of Computer Science, 1o 103-121 (1999)

Polynomial on (E,P)-free

[1305]

M.U. Gerber, V.V. Lozin
Robust algorithms for the stable set problem
Graphs and Combin., to appear

Polynomial [$O(nm)$] on (K_3,3-e,P₅,X₉₈)-free

[1117]

A. Brandstaedt, V.V. Lozin
A note on \alpha-redundant vertices in graphs.
Discrete Appl. Math. 108 (2001) 301-308

Polynomial [$O(V^{5})$] on (K_3,3-e,P₅,X₉₉)-free

[1307]

M.U. Gerber, A. Hertz, D. Schindl
P_5-free graphs and the maximum stable set problem
Discrete Appl. Math. 132 109-119 (2004)

Polynomial on (K_3,3-e,P₅)-free

[1246]

V. Alekseev, P. Lozin, R. Mosca
Maximum independent set problem and P_5-free graphs
Manuscript 2004

Polynomial on Meyniel

[169]

M. Burlet, J. Fonlupt
Polynomial algorithm to recognize a Meyniel graph
Annals of Discrete Math. 21 1984 225--252

Polynomial [$O(VE)$] on (P,P₅)-free

[1117]

A. Brandstaedt, V.V. Lozin
A note on \alpha-redundant vertices in graphs.
Discrete Appl. Math. 108 (2001) 301-308

Polynomial [$O(V^8)$] on (P,P₇)-free

[1351]

V.E. Alekseev, V.V. Lozin
Augmenting graphs for independent sets
Discrete Appl. Math. 145, No.1 3-10 (2004)

Polynomial on (P,P₈)-free

[1306]

M.U. Gerber, A. Hertz, V.V. Lozin
Stable sets in two subclasses of banner-free graphs
Discrete Appl. Math. 132 121-136 (2004)

Polynomial on (P,T₂)-free

[1305]

M.U. Gerber, V.V. Lozin
Robust algorithms for the stable set problem
Graphs and Combin., to appear

Polynomial on (P,star_1,2,5)-free

[1349]

V.L. Lozin, M. Milanic
On finding augmenting graphs
Rutcor Research Report 28-2005

Polynomial [$O(V min(d,\alpha))$] on circle

[1465]

N. Nash, D. Gregg
An output sensitive algorithm for computing a maximum independent set of a circle graph
Inform. Process. Lett. 110 No.16 630-634 (2010)

Polynomial on co-biclique separable

[1304]

Eschen, Elaine M.; Hoàng, Chính T.; Petrick, Mark D.T.; Sritharan, R
Disjoint clique cutsets in graphs without long holes
J. Graph Theory 48, No.4, 277-298 (2005)

Polynomial on comparability

[453]

M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980

Polynomial [$O(VE)$] on weakly chordal

[1119]

R. Hayward, J. Spinrad. R. Sritharan
Weakly chordal graph algorithms via handles
Proc. of the 11th symposium on Discrete Algorithms 42-49, 2000

[530]

R.B. Hayward, C. Ho\`ang, F. Maffray
Optimizing weakly triangulated graphs
Graphs and Combinatorics 5 339--349, Erratum: 6 (1990) 33--35 1989

Maximum cut

[?]

(decision variant)

Input: A graph G in this class and an integer k.

Output: True iff the vertices of G can be partitioned into two sets A,B such that there are at least k edges in G with one endpoint in A and the other endpoint in B.

Polynomial

Polynomial from XP,W-hard on booleanwidth and Polynomial decomposition time
Polynomial from XP,W-hard on cliquewidth and Linear decomposition time
Polynomial from XP,W-hard on cliquewidth and Polynomial decomposition time
Polynomial from XP,W-hard on rankwidth and Linear decomposition time

Polynomial [$O(V^2)$] on cograph

[1629]

H.L. Bodlaender, K. Jansen
On the complexity of the maximum cut problem
Nordic J. Comput. 7 No.1 14-31 (2000)

Polynomial on (q,q-4), fixed q

[1731]

H.L. Bodlaender, C.M.H. de Figueiredo, M. Gutierrez, T. Kloks, R. Niedermeier
Simple max-cut for split-indifference graphs and graphs with few P4s
Proceedings of 3rd Workshop on experimental and efficient algorithms WEA-04, LNCS 3059 87-99 (2004)

Monopolarity

[?]

Input: A graph G in this class.

Output: True iff G is monopolar.

Polynomial

Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Polynomial [$O(V^4)$] on (5-pan,T₂,X₁₇₂)-free

[1764]

V.B. Le, R. Nevries
Complexity and algorithms for recognizing polar and monopolar graphs
Theoretical Computer Science 528 1-11 (2014)

Polynomial on hole-free

[1764]

V.B. Le, R. Nevries
Complexity and algorithms for recognizing polar and monopolar graphs
Theoretical Computer Science 528 1-11 (2014)

Polynomial [$O(VE)$] on permutation

[1769]

T. Ekim, P. Heggernes, D. Meister
Polar permutation graphs are polynomial-time recognisable
European J. Combin. 34 No.3 576-592 (2013)

Polarity

[?]

Input: A graph G in this class.

Output: True iff G is polar.

Polynomial

Polynomial from Polarity on the complement
Polynomial from XP on booleanwidth and Polynomial decomposition time
Polynomial from XP on cliquewidth and Linear decomposition time
Polynomial from XP on cliquewidth and Polynomial decomposition time
Polynomial from XP on rankwidth and Linear decomposition time

Polynomial [$O(VE^2)$] on permutation

[1769]

T. Ekim, P. Heggernes, D. Meister
Polar permutation graphs are polynomial-time recognisable
European J. Combin. 34 No.3 576-592 (2013)

Recognition

[?]

Input: A graph G.

Output: True iff G is in this graph class.

Polynomial

Finite forbidden subgraph characterization

Weighted clique

[?]

Input: A graph G in this class with weight function on the vertices and a real k.

Output: True iff G contains a set S of pairwise adjacent vertices, such that the sum of the weights of the vertices in S is at least k.

Linear

Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on cliquewidth and Linear decomposition time
Polynomial from FPT on cliquewidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from Weighted independent set on the complement

Linear on comparability

[453]

M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980

Polynomial on alternation

[1598]

M.M. Halldorson, S. Kitaev, A. Pyatkin
Alternation graphs
Proceedings of WG 2011, Lecture Notes in Computer Science 6986, 191-202 (2011)

Polynomial [$O(V^2 + E \log \log V)$] on circle

[1429]

W.-L. Hsu
Maximum weight clique algorithms for circular-arc graphs and circle graphs
SIAM J. Computing 14 No.1 224-231 (1985)

Polynomial [$O(V^2 \log V)$] on circle-trapezoid
Polynomial on co-comparability ∪ comparability
Polynomial on interval filament

[1159]

F. Gavril
Maximum weight independent sets and cliques in intersection graphs of filaments.
Information Processing Letters 73(5-6) 181-188 (2000)

Polynomial [$O(VE)$] on perfectly orderable
Polynomial [$O(V log log V)$] on trapezoid

Weighted feedback vertex set

[?]

Input: A graph G in this class with weight function on the vertices and a real k.

Output: True iff G contains a set S of vertices, such that the sum of the weights of the vertices in S is at most k and every cycle in G contains a vertex from S.

Linear

Linear from FPT-Linear on cliquewidth and Linear decomposition time
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from FPT-Linear on cliquewidth and Polynomial decomposition time
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on maximum induced matching and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Polynomial on circle

[1585]

F. Gavril
Minimum weight feedback vertex sets in circle graphs
Information Proc. Lett. 107 No.1 1-6 (2008)

Polynomial [$O(V^{2n+5})$] on circle-n-gon, fixed n

Weighted independent dominating set

[?]

Input: A graph G in this class with weight function on the vertices and a real k.

Output: True iff G contains a set S of pairwise non-adjacent vertices, with the sum of the weights of the vertices in S at most k, such that every vertex in G is either in S or adjacent to a vertex in S.

Linear

Linear from FPT-Linear on cliquewidth and Linear decomposition time
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from FPT-Linear on cliquewidth and Polynomial decomposition time
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Weighted independent set

[?]

Input: A graph G in this class with weight function on the vertices and a real k.

Output: True iff G contains a set S of pairwise non-adjacent vertices, such that the sum of the weights of the vertices in S is at least k.

Linear

Linear from FPT-Linear on cliquewidth and Linear decomposition time
Polynomial from FPT on booleanwidth and Polynomial decomposition time
Polynomial from FPT on rankwidth and Linear decomposition time
Polynomial from FPT-Linear on cliquewidth and Polynomial decomposition time
Polynomial from Weighted clique on the complement
Polynomial from XP on maximum independent set and Linear decomposition time
Polynomial from XP on minimum clique cover and Linear decomposition time

Linear on (P₅,gem)-free

[1170]

H. Bodlaender, A. Brandstaedt, D. Kratsch, M. Rao, J. Spinrad
Linear time algorithms for some NP-complete problems on (P_5, gem)-free graphs.
Manuscript 2003

Linear on distance-hereditary
Linear on permutation

[1164]

V. Kamakoti, C. Pandu Rangan
Efficient Transitive Reduction on Permutation Graphs and its Applications.
CSI JL on Computer Science and Informatics, Vol. 23, No. 3, pp. 52 - 59 (1993)

Polynomial on 4K₁-free [trivial]
Polynomial on 5K₁-free [trivial]
Polynomial [$O(V^4)$] on AT-free

[160]

H. Broersma, T. Kloks, D. Kratsch, H. M\"uller
Independent sets in asteroidal triple-free graphs
SIAM J. Discrete Math. 12, No.2, 276-287 (1999)

Polynomial [$O(V^5E^3)$] on Berge ∩ bull-free

[1278]

C.M.H. de Figueiredo, F. Maffray
Optimizing bull-free perfect graphs
SIAM J. Discrete Math. Vol.18 No.2 226-240 (2004)

Polynomial on (K_1,4,P,P₅,fork)-free

[1103]

A. Brandstaedt, P.L Hammer
On the stability number of claw-free, P_5-free and more general graphs.
Rutcor Research Report 27-97 1997

Polynomial [$O(n^5)$] on (K_1,4,P₅)-free

[1110]

R. Mosca
Polynomial algorithms for the maximum stable set problem on particular classes of P_5-free graphs
Information Processing Letters 61 (1997) 137-144

Polynomial on K₂ ∪ claw-free

[1290]

V. Lozin, R. Mosca
Independent sets and extensions of 2K_2-free graphs
Discrete Appl. Math. 146 74-80 (2005)

Polynomial [$O(n^8)$] on (K_4,4,P₅)-free
Polynomial on (P,P₅)-free

[1353]

A. Brandstaedt, C.T. Hoang
On clique separators, nearly chordal graphs, and the Maximum Weight Stable Set Problem
Theoretical Comp. Sci. 389, No.1-2, 295-306 (2007)

Polynomial [$O(V^9 E)$] on (P,P₇)-free

[1452]

R. Mosca
Stable sets of maximum weight in (P_7, banner)-free graphs
Discrete Math. 308 Issue 1 20-33 (2008)

Polynomial on (P₅,X₈₂,X₈₃)-free

[1246]

V. Alekseev, P. Lozin, R. Mosca
Maximum independent set problem and P_5-free graphs
Manuscript 2004

Polynomial on (P₅,co-fork)-free

[1161]

A. Brandstaedt, R. Mosca
On the structure and stability number of P_5 and co-chair-free graphs.
To appear in Discrete Appl. Math.

Polynomial [$O(VE)$] on (P₅,fork)-free

[1125]

A. Brandstaedt, V.B. Le, H.N. de Ridder
Efficient robust algorithms for the maximum weight stable set problem in chair-free graph classes.
Universitaet Rostock, Fachbereich Informatik, Preprint CS-14-01

Polynomial on (P₅,house)-free

[1109]

V. Giakoumakis, I. Rusu
Weighted parameters in (P_5,\overline{P_5})-free graphs
Discrete Appl. Math. 80 (1997) 255-261

Polynomial on P₅-free

[1635]

D. Lokshantov, M. Vatshelle, Y. Villanger
Independent set in P5-free graphs in polynomial time
Accepted for publication

Polynomial on P₆-free

[1839]

A. Grzesik, T. Klimosova, M. Pilipczuk
Polynomial-time Algorithm for Maximum Weight Independent Set on P6-free Graphs
ACM Transactions on Algorithms 18 No.1 1-57 (2022)

Polynomial [$O(VE)$] on (P,fork)-free

[1125]

A. Brandstaedt, V.B. Le, H.N. de Ridder
Efficient robust algorithms for the maximum weight stable set problem in chair-free graph classes.
Universitaet Rostock, Fachbereich Informatik, Preprint CS-14-01

Polynomial [$O(VE)$] on (bull,fork)-free

[1124]

A. Brandstaedt, C.T. Hoang, V.B. Le
Stability number of bull- and chair-free graphs revisited
to appear in Discrete Appl. Math.

[307]

C. De Simone, A. Sassano
Stability number of bull-- and chair--free graphs
Discrete Appl. Math. 41 1993 121--129

Polynomial [$O(V^2)$] on circle

[1121]

A. Apostolico, M.J. Atallah, S.E. Hambrusch
New clique and independent set algorithms for circle graphs.
Discrete Appl. Math 36 (1992) 1-24 Erratum: Discrete Appl. Math 41 (1993) 179-180

Polynomial [$O(V^2)$] on circle-trapezoid
Polynomial [$O(V^2 \log \log V)$] on circular trapezoid
Polynomial on (co-fork,hole)-free

[1494]

A. Brandstaedt, V. Giakoumakis
Maximum Weight Independent Sets in Hole- and Co-Chair-Free Graphs
Inform. Proc. Lett. 112 67-71 (2012)

Polynomial [$O(VE)$] on co-gem-free
Polynomial [$O(V log^{d-1} V)$] on d-trapezoid
Polynomial on fork-free

[1099]

V.E. Alekseev
A polynomial algorithm for finding maximum independent sets in fork-free graphs
Discrete Ann. Operation Res., Ser. 1 6 (1999) 3-19 (in Russian)

Polynomial on interval filament

[1159]

F. Gavril
Maximum weight independent sets and cliques in intersection graphs of filaments.
Information Processing Letters 73(5-6) 181-188 (2000)

Polynomial [$O(E + V \log V)$] on multitolerance

[1497]

G.B. Mertzios
An intersection model for multitolerance graphs: Efficient algorithms and hierarchy
Proc. of 21 annual ACM-SIAM symposium on Discrete algorithms SODA2011 1306-1317 (2011)

Polynomial on (n+4)-pan-free

[1447]

A. Brandstaedt, V.V. Lozin, R. Mosca
Independent sets of maximum weight in apple-free graphs
SIAM J. Discrete Math. Vol.24 No.1 239-254 (2010)

Polynomial [$O(n^{6p+2})$] on (p,q<=2)-colorable

[1116]

V.E. Alekseev, V.V. Lozin
Independent sets of maximum weight in (p,q)-colorable graphs
Rutcor Research Report 12-2002

Polynomial on parity

[170]

M. Burlet, J.P. Uhry
Parity graphs
Annals of Discrete Math. 21 1984 253--277

Polynomial on perfect

[476]

M. Gr\"otschel, L. Lov\'asz, A. Schrijver
The ellipsoid method and its consequences in combinatorial optimization
Combinatorica 1 169--197, 1981 Corrigendum

Polynomial on semi-P₄-sparse

[402]

J.L. Fouquet, V. Giakoumakis
On semi--$P_4$--sparse graphs
Discrete Math. 165/166 1997 277--300

Polynomial on subtree overlap

[1123]

E. Cenek, L. Stewart
Maximum independent set and maximum clique algorithms for overlap graphs
Discrete Appl. Math. 131, No.1 77-91 (2003)

Polynomial [$O(V \log V)$] on tolerance
Polynomial [$O(V^2)$] on tolerance

[1498]

G.B. Mertzios, I. Sau, S. Zaks
A new intersection model and improved algorithms for tolerance graphs
SIAM J. on Discrete Math. 23(4) 1800-1813 (2009)

Polynomial [$O(V \log \log V)$] on trapezoid
Polynomial [$O(V^4)$] on weakly chordal

[997]

J.P. Spinrad, R. Sritharan
Algorithms for weakly triangulated graphs
Discrete Appl. Math. 59 1995 181--191

Weighted maximum cut

[?]

(decision variant)

Input: A graph G in this class with weight function on the edges and a real k.

Output: True iff the vertices of G can be partitioned into two sets A,B such that the sum of weights of the edges in G with one endpoint in A and the other endpoint in B is at least k.

NP-complete

NP-complete on 2K₁-free

[1676]

M. Kaminski
Max-cut and containment relations in graphs
Proceedings of WG 2010, Lecture Notes in Computer Science 6410, 15-26 (2010)

Graphclass: (4K₁,P₄)-free

Complement classes

Forbidden subgraphs

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems

Graphclass: (4K1,P4)-free

Complement classes

Forbidden subgraphs

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems

Graphclass: (4K₁,P₄)-free