Graphclass: pseudo-split

The following definitions are equivalent:

A graph is pseudo-split if its vertex set can be partitioned into three possibly empty sets C, S and I, such that
- C is a complete graph, I is independent and S (if non-empty) induces a C₅ ;
- every vertex in C is adjacent to every vertex in S;
- every vertex in I is non-adjacent to every vertex in S.
A graph is called pseudo-split if it is (2K₂,C₄)-free .

References

[758]

F. Maffray, M. Preissmann
Linear recognition of pseudo--split graphs
Discrete Appl. Math. 52 1994 307--312

;

[1808]

(no preview available)

Equivalent classes

Only references for direct inclusions are given. Where no reference is given for an equivalent class, check other equivalent classes or use the Java application.

(2K₂,C₄)-free
[1169]
(no preview available)
β-perfect ∩ co-β-perfect

Complement classes

self-complementary

Inclusions

The map shows the inclusions between the current class and a fixed set of landmark classes. Minimal/maximal is with respect to the contents of ISGCI. Only references for direct inclusions are given. Where no reference is given, check equivalent classes or use the Java application. To check relations other than inclusion (e.g. disjointness) use the Java application, as well.

Map

Inclusion map for pseudo--split

Minimal superclasses

(1,2)-colorable
[1751]
(no preview available)

(known proper)
(2,1)-colorable (known proper)
(2K₂,2P₃,C₆)-free (known proper)
(2K₂,house)-free (known proper)
(2K₃ + e,5-pan,A,C₆,E,K_3,3-e,P₆,R,X₁₆₆,X₁₆₇,X₁₆₉,X₁₇₀,X₁₇₁,X₁₇₂,X₁₈,X₃₇,X₄₅,X₅,X₅₈,X₈₄,X₉₅,X₉₈,5-pan,A,C₆,E,P₆,R,X₁₆₆,X₁₆₇,X₁₆₉,X₁₇₀,X₁₇₁,X₁₇₂,X₁₈,X₃₇,X₄₅,X₅,X₅₈,X₈₄,X₉₅,X₉₈,antenna,co-antenna,co-domino,co-fish,co-twin-C₅,co-twin-house,domino,fish,twin-C₅,twin-house)-free (known proper)
(2P₃,C₄,C₆)-free (known proper)
(C₄,P₅)-free (known proper)
β-perfect (known proper)
circle-polygon (known proper)
co-β-perfect (known proper)
co-interval filament (known proper)
co-interval mixed (known proper)
extended P₄-laden
[1169]
(no preview available)

(known proper)
polar
[1808]
(no preview available)

(known proper)
spider graph
[1752]
(no preview available)

(known proper)

Maximal subclasses

(1,1)-colorable (known proper)
(2K₂,C₄,C₅)-free (known proper)
chordal ∩ co-chordal (known proper)
split (known proper)

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)}$ )

Superfactorial (2ドル^{\Theta(n^2)}$)

at least Superfactorial (2ドル^{\Theta(n^2)}$) on split

[1788]

V.E. Alekseev
Range of values of entropy of hereditary classes of graphs
Diskretn. Math. 4:2 148-157 (1995)

[1789]

B. Bollobas, A. Thomason
Projections of bodies and hereditary properties of hypergraphs
Bull. London Math. Soc. 27(5) 417-424 (1995)

Parameters

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from cutwidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from rankwidth
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.

Unbounded

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

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from rankwidth
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 $j$.

Unbounded

Unbounded From Superfactorial(Theta) Speed.
Unbounded From Superfactorial(Theta) Speed.
Unbounded from Domination assuming Linear,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from booleanwidth
Unbounded from rankwidth

Unbounded on permutation ∩ split

[1796]

N. Korpelainen, V.V. Lozin, C. Mayhill
Split permutation graphs
Graphs and Combinatorics 30 633-646 (2014)

Unbounded on split

[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

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 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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from rankwidth
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 Superfactorial(Theta) Speed.
Unbounded From Superfactorial(Theta) Speed.
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 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 from Maximum cut assuming Polynomial,NP-complete disjoint.

Unbounded on complete split [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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
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 Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from distance to block
Unbounded from distance to cograph

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
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

Unbounded from distance to cograph on the complement

Unbounded on (2K₂,C₄,C₅,claw,diamond)-free

[1866]

(no preview available)

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from pathwidth
Unbounded from rankwidth
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

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from bandwidth
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from chromatic number
Unbounded from cliquewidth
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 rankwidth
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]
Unbounded on hamiltonian ∩ split [trivial]

maximum degree

[?]

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

Unbounded

Unbounded From Superfactorial(Theta) Speed.
Unbounded From Superfactorial(Theta) Speed.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from chromatic number
Unbounded from degeneracy
Unbounded from maximum clique

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

Unbounded

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
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 clique
Unbounded from pathwidth
Unbounded from rankwidth
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

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 from Domination assuming Polynomial,NP-complete disjoint.

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

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 $G$.

Unbounded

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from pathwidth
Unbounded from rankwidth
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 Domination assuming Linear,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Linear,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Linear,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from maximum clique
Unbounded from rankwidth

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 Domination assuming Polynomial,NP-complete disjoint.
Unbounded from Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from Hamiltonian cycle assuming Polynomial,NP-complete disjoint.
Unbounded from Hamiltonian path assuming Polynomial,NP-complete disjoint.
Unbounded from Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from chromatic number
Unbounded from cliquewidth
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 clique
Unbounded from maximum matching
Unbounded from pathwidth
Unbounded from rankwidth
Unbounded from tree depth
Unbounded from treewidth

Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the current class.

Parameter decomposition

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.

Unknown to ISGCI

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.

Unknown to ISGCI

cutwidth decomposition

[?]

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

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

NP-complete

Unbounded/NP-complete on split

[1510]

P. Heggernes, D. Lokshtanov, R. Mihai, C. Papadopoulos
Cutwidth of split graphs, threshold graphs, and proper interval graphs
Proceedings of WG 2008, LNCS 5344, pp. 218-229 (2008)

treewidth decomposition

[?]

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

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

Unknown to ISGCI

Unweighted problems

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.

Polynomial

Polynomial from Colourability

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 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)

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.

Polynomial

Polynomial from Independent set on the complement
Polynomial from Weighted clique

Polynomial on b-perfect

[1478]

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

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

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.

Polynomial

Polynomial on β-perfect

[771]

S.E. Markosjan, G.S. Gasparian, B. Reed
$\beta$--perfect graphs
J. Comb. Theory (B) 67 1996 1--11

Polynomial on b-perfect

[1478]

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

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.

NP-complete

NP-complete on split

[1144]

A.A. Bertossi
Dominating sets for split and bipartite graphs.
Inform. Process. Lett. 19 (1984) 37-40

[1145]

D.G. Corneil, Y. Perl
Clustering and domination in perfect graphs.
Discrete. Appl. Math. 9 (1984) 27-39

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.

Polynomial

Polynomial from Weighted feedback vertex set

Graph isomorphism

[?]

Input: Graphs G and H in this class

Output: True iff G and H are isomorphic.

GI-complete

GI-complete on split

[1695]

F.R.K. Chung
On the cutwidth and the topological bandwidth of a tree
SIAM J. Alg. Discr. Meth. 6 1985 268--277

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.

NP-complete

NP-complete on split

[453]

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

NP-complete on split ∩ strongly chordal

[1517]

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

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.

NP-complete

NP-complete on split ∩ strongly chordal

[1517]

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

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 Weighted independent set

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

Polynomial on (C₄,P₆)-free

[1351]

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

[1352]

R. Mosca
Independent sets in certain P_6-free graphs
Discrete Applied Math. 92 177-191 (1999)

Polynomial on (C₆,K_3,3+e,P,P₇,X₃₇,X₄₁)-free

[1346]

N.V.R. Mahadev, B.A. Reed
A note on vertex orders for stability number
J. Graph Theory 30 113-120 (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 on (K_2,3,P,P₅)-free

[1107]

N.V.R. Mahadev
Vertex deletion and stability number
Research report ORWP 90/2 Dept. of Mathematics, Swiss Fed.Inst. of Technology 1990

[1346]

N.V.R. Mahadev, B.A. Reed
A note on vertex orders for stability number
J. Graph Theory 30 113-120 (1999)

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 [$O(VE)$] 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 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)

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.

NP-complete

NP-complete on split

[1629]

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

Monopolarity

[?]

Input: A graph G in this class.

Output: True iff G is monopolar.

Linear

[1808]

(no preview available)

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)

Polarity

[?]

Input: A graph G in this class.

Output: True iff G is polar.

Linear

Linear on polar [trivial]

Recognition

[?]

Input: A graph G.

Output: True iff G is in this graph class.

Linear

Linear on (2K₂,C₄)-free

[758]

F. Maffray, M. Preissmann
Linear recognition of pseudo--split graphs
Discrete Appl. Math. 52 1994 307--312

Polynomial on (2K₂,C₄)-free

Weighted problems

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.

Polynomial

Polynomial from Weighted independent set on the complement

Polynomial [$O(V^2 E)$] on C₄-free ∩ odd-signable

[1476]

M.V. da Silva, K. Vuskovic
Triangulated neighborhoods in even-hole-free graphs
Discrete Math. 307 1065-1073 (2007)

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)

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.

Polynomial

Polynomial from XP on maximum induced matching 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

Polynomial from Weighted clique on the complement

Linear on (2K₂,C₄)-free
Polynomial on 2K₂-free

[1160]

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

Polynomial [$O(n^4)$] on (C₄,P₅)-free
Polynomial on (C₄,P₆)-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 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 (K_2,3,P,P₅)-free

[1107]

N.V.R. Mahadev
Vertex deletion and stability number
Research report ORWP 90/2 Dept. of Mathematics, Swiss Fed.Inst. of Technology 1990

Polynomial on (K_2,3,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 [$O(n^6)$] on (K_3,3,P₅)-free
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,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 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 (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 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 [$O(n^{6p+2})$] 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 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)

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 from Maximum cut

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: pseudo-split

References

Equivalent classes

Complement classes

Related classes

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems