2K_2--free $\cap$ bipartite 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 (2K₂,C₅,triangle)-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 degeneracy

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 cutwidth
Unbounded from degeneracy
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from treewidth

Unbounded on complete bipartite [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 degeneracy

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 degeneracy
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 degeneracy
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.

Bounded

Bounded on 4-colorable [by definition]
Bounded on 5-colorable [by definition]
Bounded on 6-colorable [by definition]
Bounded on bipartite [by definition]
Bounded on tripartite [by definition]

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 (K₂ ∪ claw,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

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,diamond)-free

[1190]

A. Brandstaedt
(P_5,diamond)-free graphs revisited: Structure and linear time optimization.
Accepted for Discrete Appl. Math.

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 (P₆,triangle)-free

[1435]

A. Brandstaedt, K. Klembt, S. Mahfud
P_6- and triangle-free graphs revisited: structure and bounded clique-width
Discrete Math. Theor. Comput. Sci. 8 173-187 (2006)

Bounded on (P₇,odd-cycle,star_1,2,3)-free

[1181]

V. Giakoumakis, J.M. Vanherpe
Linear time recognition and optimization for weak-bisplit graphs, bi-cographs and bipartite P_6-free graphs.
Manuscript, 2001

Bounded on (X₁₇₂,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

Bounded on (X₁₇₇,odd-cycle)-free

[1437]

V. Lozin, J. Volz
The clique-width of bipartite graphs in monogenic classes
International J. of Foundations of Comp. Sci. 19 477-494 (2008)

Bounded on cliquewidth 3
Bounded on cliquewidth 4
Bounded on difference

[1192]

V. Lozin, D. Rautenbach
Chordal bipartite graphs of bounded tree- and clique-width.
Rutcor Research Report 2-2003

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 (odd-cycle,star_1,2,3,sunlet₄)-free

[1139]

Lozin, Vadim V.
On a generalization of bi-complement reducible graphs.
Proceedings of MFCS 2000, Lect. Notes Comput. Sci. 1893, 528-538 (2000)

Bounded on (odd-cycle,star_1,2,3)-free

[1140]

Lozin, Vadim V.
Bipartite graphs without a skew star
Rutcor Research Report RRR 20-2001

Bounded on probe P₄-reducible
Bounded on probe P₄-sparse
Bounded on probe bipartite chain
Bounded on probe bipartite distance-hereditary
Bounded on probe distance-hereditary
Bounded on probe ptolemaic
Bounded on semi-P₄-sparse

[1186]

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

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 chromatic number

Bounded on bipartite ∪ co-bipartite ∪ split [trivial]
Bounded on (p,q<=2)-colorable

[1866]

(no preview available)

Bounded on probe (2,2)-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 degeneracy
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 on complete bipartite [by definition]

diameter

[?]

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

Bounded

Bounded from maximum induced matching

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 complete bipartite [trivial]

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
Unbounded from distance to cograph
Unbounded from maximum independent set
Unbounded from minimum clique cover
Unbounded from minimum dominating set

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
Unbounded from distance to cograph

Unbounded on complete bipartite [by definition]

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 cograph

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 .

Unbounded

Take a P₄ and add many false twins to each vertex.

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 degeneracy
Unbounded from distance to block
Unbounded from distance to outerplanar
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 degeneracy
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 degeneracy

Unbounded on complete bipartite [by definition]

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 cutwidth
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
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 $G$.

Bounded

Bounded from chromatic number

Bounded on K₄-free [by definition]
Bounded on K₆-free [by definition]
Bounded on K₇-free [by definition]
Bounded on triangle-free [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 degeneracy

Unbounded on (C₄,P₃,triangle)-free [by definition]
Unbounded on complete bipartite [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 $G$.

Unbounded

Unbounded from minimum dominating set

Unbounded on complete bipartite [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 on 2K₂-free [by definition]

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 degeneracy
Unbounded from distance to block
Unbounded from distance to cluster
Unbounded from distance to co-cluster
Unbounded from distance to cograph
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
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.

Unbounded

Unbounded from maximum independent set
Unbounded from minimum dominating set

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 $G$.

Unbounded

Unbounded on K₂-free [by definition]

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 degeneracy
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 degeneracy
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 degeneracy

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 degeneracy
Unbounded from distance to block
Unbounded from distance to cluster
Unbounded from distance to co-cluster
Unbounded from distance to cograph
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
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 Dilworth 4
Polynomial on circular permutation
Polynomial on circular trapezoid
Polynomial on convex
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 (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,diamond)-free

[1190]

A. Brandstaedt
(P_5,diamond)-free graphs revisited: Structure and linear time optimization.
Accepted for Discrete Appl. Math.

Linear on (P₇,odd-cycle,star_1,2,3)-free

[1181]

V. Giakoumakis, J.M. Vanherpe
Linear time recognition and optimization for weak-bisplit graphs, bi-cographs and bipartite P_6-free graphs.
Manuscript, 2001

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 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 [$O(V^2+VE)$] on (odd-cycle,star_1,2,3,sunlet₄)-free

[1139]

Lozin, Vadim V.
On a generalization of bi-complement reducible graphs.
Proceedings of MFCS 2000, Lect. Notes Comput. Sci. 1893, 528-538 (2000)

Polynomial [$O(V^2+VE)$] on (odd-cycle,star_1,2,3)-free

[1140]

Lozin, Vadim V.
Bipartite graphs without a skew star
Rutcor Research Report RRR 20-2001

cutwidth decomposition

[?]

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

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

Linear

Linear on bipartite permutation

[1507]

P. Heggernes, P. van 't Hof, D. Lokshtanov, J. Nederlof
Computing the cutwidth of bipartite permutation graphs in linear time
Proceedings of the 36th international conference on Graph-theoretic concepts in computer science WG 2010 75-87 (2010)

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 chordal bipartite

[119]

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

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-chordal

[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 co-interval

[1416]

R. Garbe
Tree-width and path-width of comparability graphs of interval orders
20th Intern. Workshop on Graph--Theoretic Concepts in Comp. Sci. WG'94, Lecture Notes in Comp. Sci. 903 (1995) 26-37

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 [$O(V^2)$] 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 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
Polynomial from Colourability
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 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 [$O(V^4)$] 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 nK₂-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)

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (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
Polynomial from XP on chromatic number and Linear decomposition time
Polynomial from XP on maximum clique 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

Polynomial on B₀-VPG
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 [$O(V \log V)$] on boxicity 2

[604]

H. Imai, T. Asano
Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane
J. Algorithms 4 1983 310--323

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 [$O(V^{2.5}/\log V)$] on generalized split
Polynomial on locally chordal
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 [$O(V+E)$] on generalized split

[1798]

E.M. Eschen, X. Wang
Algorithms for unipolar and generalized split graphs
Discrete Applied Math. 162 195-201 (2014)

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 (0,3)-colorable
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 bipartite [trivial]
Linear on paw-free ∩ perfect
Polynomial on (2P₃,triangle)-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 (3K₂,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

Polynomial on (K₂ ∪ claw,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

Polynomial on (P₂ ∪ P₄,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

Polynomial on (P₆,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

Polynomial on (X₁₇₂,triangle)-free

[1434]

K. Dabrowski, V. Lozin, R. Raman, B. Ries
Colouring vertices of triangle-free graphs
Proceedings of the 36th International Workshop on Graph-Theoretic Concepts in Computer Science WG 2010, LNCS 6410, 184-195 (2010)

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 on clique separable

[1424]

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

Polynomial [$O(V^3)$] 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 [$O(V^{2.5}/\log V)$] on generalized split
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

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 on (K₄,P₅)-free

[1257]

I.E. Zverovich
The domination number of (K_p,P_5)-free graphs
Australas. J. Comb. 27 95-100 (2003)

Polynomial on (P₅,triangle)-free

[1257]

I.E. Zverovich
The domination number of (K_p,P_5)-free graphs
Australas. J. Comb. 27 95-100 (2003)

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 co-interval ∪ interval
Polynomial on convex

[1167]

Damaschke, Peter; Müller, Haiko; Kratsch, Dieter
Domination in convex and chordal bipartite graphs.
Inf. Process. Lett. 36, No.5, 231-236 (1990)

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 probe interval

[1340]

T. Kloks, C.-S. Liu, S.-L. Peng
Domination and independent domination on probe interval graphs
Proceedings of 23rd Workshop on Combinatorial Mathematics and Computation Theory 93-97 (2006)

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

Linear on bipartite permutation

[1584]

A. Takaoka, S. Tayu, S. Ueno
On minimum feedback vertex set in graphs
Third Internation Conference on Networking and Computing 429-434 (2012)

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 on chordal ∪ co-chordal
Polynomial on chordal bipartite

[1583]

T. Kloks, C.-H. Liu, S.-H. Poon
Feedback vertex set on chordal bipartite graphs
Manuscript (2012)

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 on co-interval ∪ interval
Polynomial [$O(V^2E)$] on convex

[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 on leaf power ∪ min leaf power
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 co-interval ∪ interval

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 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 convex

[1517]

H. Mueller
Hamiltonian circuits in chordal bipartite graphs
Discrete Math. 156 291-298 (1996)

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 bipartite 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 circular convex bipartite

[1245]

Y. Daniel Liang, N. Blum
Circular convex bipartite graphs: Maximum matching and Hamiltonian circuits
Inform. Proc. Letters Vol.56 No.4 215-219 (1995)

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 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 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 bipartite 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 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

Linear on co-chordal

[558]

C.T. Ho\`ang
Recognition and optimization algorithms for co--triangulated graphs
Forschungsinstitut f\"ur Diskrete Mathematik, Bonn, 1990 Tech. Report No. 90637-OR

Linear on co-comparability

[1100]

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

Polynomial on (C₅,P₅,P₂ ∪ P₃)-free

[1118]

M.U. Gerber, V.V. Lozin
On the stable set problem in special P_5-free graphs.
Rutcor Research Report 24-2000

Polynomial on Gallai

[1081]

S.H. Whitesides
A method for solving certain graph recognition and optimization problems, with applications to perfect graphs
Annals of Discrete Math. 21 1984 281--297

Polynomial on Meyniel

[169]

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

Polynomial on (P₅,P₂ ∪ P₃)-free

[1350]

V.E. Alekseev
On easy and hard hereditary classes of graphs with respect to the independent set problem
Discrete Appl. Math. 132, No.1-3 17-26 (2003)

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 clique separable

[1081]

S.H. Whitesides
A method for solving certain graph recognition and optimization problems, with applications to perfect graphs
Annals of Discrete Math. 21 1984 281--297

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 co-hereditary clique-Helly

[1298]

E. Prisner
Hereditary clique-helly graphs
J. Comb. Math. Comb. Comput 14 216-220 (1993)

Polynomial on comparability

[453]

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

Polynomial [$O(V+E)$] on generalized split

[1798]

E.M. Eschen, X. Wang
Algorithms for unipolar and generalized split graphs
Discrete Applied Math. 162 195-201 (2014)

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.

Linear

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

Linear on bipartite

[1629]

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

[trivial]

Monopolarity

[?]

Input: A graph G in this class.

Output: True iff G is monopolar.

Linear

Linear on monopolar [trivial]
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 bipartite ∪ co-bipartite ∪ co-line graphs of bipartite graphs ∪ line graphs of bipartite graphs
Polynomial on bipartite ∪ co-bipartite ∪ split
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.

Linear

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

Linear on polar [trivial]
Polynomial on bipartite ∪ co-bipartite ∪ co-line graphs of bipartite graphs ∪ line graphs of bipartite graphs
Polynomial on chordal ∪ co-chordal
Polynomial on co-interval ∪ interval
Polynomial on leaf power ∪ min leaf power
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.

Linear

Linear on bipartite chain

[1468]

P. Heggernes, D. Kratsch
Linear-time certifying recognition algorithms and forbidden induced subgraphs
Nordic Journal of Computing 14 87-108 (2007)

Polynomial on (2K₂,C₅,triangle)-free

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
Polynomial from XP on chromatic number and Linear decomposition time
Polynomial from XP on maximum clique and Linear decomposition time

Linear on comparability

[453]

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

Polynomial on 4-colorable [trivial]
Polynomial on 5-colorable [trivial]
Polynomial on 6-colorable [trivial]
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 on bipartite ∪ co-bipartite ∪ split
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 on maximal clique irreducible

[1642]

(no preview available)

Polynomial on monopolar

[1832]

M. Barbato, D. Bezzi
Monopolar graphs: Complexity of computing classical graph parameters
Discrete Appl. Math. 291 277-285 (2021)

Polynomial [$O(VE)$] on perfectly orderable
Polynomial [$O(VE)$] on split-neighbourhood

[759]

F. Maffray, M. Preissmann
Split--neighbourhood graphs and the strong perfect graph conjecture
J. Comb. Theory (B) 63 1995 294--309

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 induced matching 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
Polynomial on co-interval ∪ interval

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

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

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 P₅-free ∩ tripartite

[1493]

F. Maffray, G. Morel
On 3-colorable P5-free graphs
Les Cahiers Leibniz No. 191

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 2K₂-free

[1160]

M. Farber
On diameters and radii of bridged graphs.
Discrete Math. 73 (1989) 249-260

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 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 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 on (P₅,cricket)-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 (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 on bipartite
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 [$O(V log^{d-1} V)$] on d-trapezoid
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 nK₂-free, fixed n

[1102]

E. Balas, C.S.Yu
On graphs with polynomially solvable maximum weight clique problem
Networks 19 (1989) 247-253

Polynomial on nearly bipartite
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

Graphclass: 2K₂-free ∩ bipartite

Equivalent classes

Complement classes

Related classes

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems

Graphclass: 2K2-free ∩ bipartite

Equivalent classes

Complement classes

Related classes

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems

Graphclass: 2K₂-free ∩ bipartite