modular graphs

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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from cliquewidth
Unbounded from cutwidth
Unbounded from degeneracy
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from rankwidth
Unbounded from treewidth

Unbounded on binary tree ∩ partial grid

[1757]

(no preview available)

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.

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 cliquewidth
Unbounded 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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from rankwidth
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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
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.

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 $j$.

Unbounded

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 booleanwidth
Unbounded from rankwidth

Unbounded on bipartite permutation

[1182]

A. Brandstaedt, V.V. Lozin
On the linear structure and clique-width of bipartite permutation graphs.
Accepted for Ars Combinatorica

Unbounded on grid

[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)

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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
Unbounded from cliquewidth
Unbounded from degeneracy
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 on complete bipartite [by definition]
Unbounded on hypercube [by definition]

diameter

[?]

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

Unbounded

Unbounded on binary tree ∩ partial grid [trivial]
Unbounded on caterpillar [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 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 diameter
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 maximum induced matching
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 diameter
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 diameter
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 diameter

Unbounded on 2K₂-free ∩ bipartite

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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to outerplanar
Unbounded from pathwidth
Unbounded from rankwidth
Unbounded from treewidth

Unbounded on binary tree ∩ partial grid
Unbounded on caterpillar [trivial]

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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from rankwidth
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 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 acyclic chromatic number
Unbounded from bandwidth
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from carvingwidth
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 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 $G$.

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 Graph isomorphism assuming Polynomial,GI-complete disjoint.
Unbounded from acyclic chromatic number
Unbounded from degeneracy

Unbounded on (C₄,P₃,triangle)-free [by definition]
Unbounded on caterpillar [by definition]
Unbounded on complete bipartite [by definition]
Unbounded on disjoint union of stars [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 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 diameter
Unbounded from maximum induced matching
Unbounded from minimum dominating set

Unbounded on complete bipartite [by definition]
Unbounded on disjoint union of stars [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 $G$.

Unbounded

Unbounded from diameter

Unbounded on (P₄,triangle)-free [by definition]
Unbounded on maximum degree 1 [trivial]

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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from diameter
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 induced matching
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

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 diameter
Unbounded from maximum independent set
Unbounded from maximum induced matching
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 from Domination assuming Polynomial,NP-complete disjoint.
Unbounded from diameter

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 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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from rankwidth
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 $G$.

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 booleanwidth
Unbounded from cliquewidth

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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from diameter
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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from rankwidth

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 acyclic chromatic number
Unbounded from book thickness
Unbounded from booleanwidth
Unbounded from branchwidth
Unbounded from cliquewidth
Unbounded from degeneracy
Unbounded from diameter
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 induced matching
Unbounded from maximum matching
Unbounded from pathwidth
Unbounded from rankwidth
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.

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.

Unknown to ISGCI

treewidth decomposition

[?]

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

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

Unknown to ISGCI

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
Polynomial from Colourability

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 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

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

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

[1798]

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

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

Linear on (0,3)-colorable
Linear on bipartite [trivial]
Linear on paw-free ∩ perfect
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 [$O(V^2)$] 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 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

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 chordal bipartite

[1156]

Mueller, Haiko; Brandstaedt, Andreas
The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs.
Theor. Comput. Sci. 53, 257-265 (1987)

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.

Unknown to ISGCI

Graph isomorphism

[?]

Input: Graphs G and H in this class

Output: True iff G and H are isomorphic.

GI-complete

GI-complete on chordal bipartite

[1689]

R. Uehara, S. Toda, T. Nagoya
Graph isomorphism completenes for chordal bipartite graphs and strongly chordal graphs
Discrete Appl. Math. 145 No.3 479-482

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 chordal bipartite

[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 chordal bipartite

[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.

Polynomial

Polynomial from Clique on the complement
Polynomial from Weighted independent set

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 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 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)

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

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 on bipartite ∪ co-bipartite ∪ co-line graphs of bipartite graphs ∪ line graphs of bipartite graphs
Polynomial on bipartite ∪ co-bipartite ∪ split

Polarity

[?]

Input: A graph G in this class.

Output: True iff G is polar.

Linear

Linear on polar [trivial]
Polynomial on bipartite ∪ co-bipartite ∪ co-line graphs of bipartite graphs ∪ line graphs of bipartite graphs

Recognition

[?]

Input: A graph G.

Output: True iff G is in this graph class.

Polynomial

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 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 on co-comparability ∪ comparability
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

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.

Unknown to ISGCI

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.

Polynomial

Polynomial from Weighted clique on the complement

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 bipartite
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

Graphclass: modular

References

Equivalent classes

Related classes

Inclusions

Map

Minimal superclasses

Maximal subclasses

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems