Graphclass: chordal

The following definitions are equivalent:

A graph G is chordal if every minimal cutset in every induced subgraph of G is a clique.
A graph G is chordal if it is the intersection graph of subtrees of a tree T. In particular T can be chosen such that each node of T corresponds to a maximal clique of G and the subtrees T_v consist of precisely those maximal cliques in G that contain v. T is then called the clique tree of G.
A graph is chordal if every cycle of length at least 4 has a chord.
A vertex v of G is called simplicial in G if N(v) is a clique in G.
The ordering v₁ ...v_n of the vertices of G is a perfect elimination order of G if for all i, v_i is simplicial in G[v₁ ...v_i ].
A graph is chordal if it has a perfect elimination order.
Let $G$ be a graph. A tree-layout $T$ of $G$ is a rooted tree on the vertices of $G,ドル such that for every edge $xy\in E(G),ドル either $x$ is an ancestor of $y$ in $T,ドル denoted by $x\prec_{T} y,ドル or $y$ is an ancestor of $x$ in $T$.
Let $H$ be a graph with a total ordering $\prec_H$ on its vertices and let $G$ be a graph with (partial) vertex ordering $\prec_G$. We say $(H,\prec_H)$ is a pattern of $(G,\prec_G)$ iff $H$ is an induced subgraph of $G$ and for every pair of vertices $x,y$ of $H,ドル $x\prec_H y \Leftrightarrow x\prec_G y$. We encode a pattern by listing its set of (ordered) edges and non-edges.
A graph $G$ is chordal if it has a tree-layout $T$ such that $(G,\prec_T)$ does not contain any of the patterns $\langle\overline{12},13,\overline{23}\rangle, \langle\overline{12},13,23\rangle$.

References

[453]

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

[501]

A. Hajnal, J. Sur\'anyi
\"Uber die Aufl\"osung von Graphen in vollst\"andige Teilgraphen
{\sl Ann. Univ. Sci. Budapest, E\"otv\"os Sect. Math. 1} 1958 113--121

;

[119]

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

[425]

F. Gavril
The intersection graphs of subtrees in trees are exactly the chordal graphs
J. Comb. Theory (B) 16 1974 47--56

[931]

D.J. Rose, R.E. Tarjan, G.S. Lueker
Algorithmic aspects of vertex elimination on graph
SIAM J. Computing 5 1976 266--283

[1020]

R.E. Tarjan, M. Yannakakis
Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs.
SIAM J. Comput. 13, 566-579 (1984). [ISSN 0097-5397]

[1146]

K.S. Booth, J.H. Johnson
Dominating sets in chordal graphs
SIAM J. Comput. 11 (1982) 191-199

[1166]

A. Frank
Some polynomial algorithms for certain graphs and hypergraphs.
Proc. 5th Br. comb. Conf., Aberdeen 1975, Congr. Numer. XV, 211-226 (1976).

[1424]

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

[1525]

C.J. Colbourn, L.K. Stewart
Dominating cycles in series-parallel graphs
Ars Combin. 19A 107-112 (1985)

[1574]

P. Festa, P.M. Pardalos, M.G.C. Resende
Feedback set problems
in: D.Z. Du, P.M. Pardalos, Handbook of Combinatorial Optimization, Supplement vol. A, Kluwer Academic Publishers, 209-259 (2000)

[1629]

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

[1669]

M. Farber
Independent domination in chordal graphs
Operations Research Lett. 1 No.4 134-138 (1982)

[1688]

V.N. Zemlyachenko, N.M. Korneenko, R.I. Tyshkevich
Graph isomorphism problem
J. of Soviet Mathematics 29 No.4 1426-1481

[1767]

T. Ekim, P. Hell, J. Stacho, D. de Werra
Polarity of chordal graphs
Discrete Applied Mathematics 156 No. 13 2469-2479 (2008)

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.

C_n+4-free [by definition]
β-perfect ∩ perfect
[1382]
A. Brandstaedt, V.B. Le, J. Spinrad
Graph classes: a survey
SIAM Monographs on discrete mathematics and applications (1999)

[771]
S.E. Markosjan, G.S. Gasparian, B. Reed
$\beta$--perfect graphs
J. Comb. Theory (B) 67 1996 1--11
rigid circuit
[1382]
A. Brandstaedt, V.B. Le, J. Spinrad
Graph classes: a survey
SIAM Monographs on discrete mathematics and applications (1999)

[317]
G. Dirac
On rigid circuit graphs
Abhandl. Math. Seminar Univ. Hamburg 25 1961 71--76
triangulated [by definition]

Complement classes

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 chordal

Minimal superclasses

(5,2)-odd-noncrossing-chordal (known proper)
(C₄,C₅,C₆,C₇,C₈)-free (known proper)
(C₄,C₅)-free ∩ cop-win (possibly equal)
(C₄,odd-hole)-free (known proper)
C₄-free ∩ perfect (known proper)
Gallai [by definition] (known proper)
HHP-free (known proper)
Helly 2-acyclic subtree (known proper)
(K_2,3,P,hole)-free (known proper)
W_n+4-free (known proper)
β-perfect (known proper)
alternately colourable
[556]
C.T. Ho\`ang
Alternating orientation and alternating colouration of perfect graphs
J. Comb. Theory (B) 42 1987 264--273

(known proper)
bridged [by definition] (possibly equal)
building-free ∩ odd-signable (known proper)
chordal ∪ co-chordal [trivial] (known proper)
circle-polygon
[600]
C. Hundack, H. Stamm--Wilbrandt
Extended circle graphs I
Tech. Report IAI-TR-95-5, Rheinische Friedrich-Wilhelm-Universit\"at Bonn, Institut f\"ur Informatik III 0

(known proper)
even-hole-free ∩ probe chordal (known proper)
hereditary dismantlable (possibly equal)
i-triangulated (known proper)
locally chordal (known proper)
neighbourhood chordal (known proper)
perfect ∩ split-neighbourhood
[759]
F. Maffray, M. Preissmann
Split--neighbourhood graphs and the strong perfect graph conjecture
J. Comb. Theory (B) 63 1995 294--309

(known proper)
probe chordal ∩ weakly chordal (known proper)
slightly triangulated
[763]
F. Maire
Slightly triangulated graphs are perfect
Graphs and Combinatorics 10 1994 263--268

(known proper)
spider graph (known proper)

Maximal subclasses

(2,2)-colorable ∩ chordal (known proper)
3-tree [by definition] (known proper)
(3K₃,C_n+4)-free (known proper)
(C_n+4,H)-free (known proper)
(C_n+4,T₂,net)-free (known proper)
(C_n+4,X₅₉,longhorn)-free (known proper)
Helly chordal (possibly equal)
chordal ∩ clique-Helly [trivial] (possibly equal)
chordal ∩ clique-chordal [trivial] (possibly equal)
chordal ∩ diametral path (known proper)
chordal ∩ domination perfect (known proper)
chordal ∩ hamiltonian [trivial] (possibly equal)
chordal ∩ irredundance perfect (known proper)
chordal ∩ planar [trivial] (possibly equal)
chordal ∩ unipolar [trivial] (possibly equal)
k-tree, fixed k
[453]
M.C. Golumbic
Algorithmic Graph Theory and Perfect Graphs
Academic Press, New York 1980

(known proper)
power-chordal (possibly equal)
substar [by definition] (known proper)
undirected path [by definition]
[175]
K.B. Cameron
Polyhedral and Algorithmic Ramifications of Antichains
Ph. D. Thesis, {\sl University of Waterloo, Canada} 1982

(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 factorial on P₃-free

[1791]

V.V. Lozin, C. Mayhill, V. Zamaraev
Locally bounded coverings and factorial properties of graphs
European J. Combin. 33 No.4 534-543 (2012)

[1792]

(no preview available)

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 cochromatic 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 cochromatic number
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 binary tree ∩ partial grid

[1757]

(no preview available)

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 cochromatic 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

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 cochromatic number
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

[1174]

B. Courcelle, S. Olariu
Upper bounds to the clique-width of graphs.
Discrete Appl. Math. 101 (2000) 77-114

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

Unbounded on unit interval

[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 $\overline{G}$.

Unbounded

Unbounded on cluster

[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 cochromatic number
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 cochromatic 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 $G$.

Unbounded

Unbounded on SC 2-tree [trivial]
Unbounded on binary tree ∩ partial grid [trivial]
Unbounded on caterpillar [by definition]
Unbounded on linear forest [by definition]
Unbounded on unit interval [trivial]

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 2-tree
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 cochromatic number
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 Maximum cut assuming Polynomial,NP-complete disjoint.
Unbounded from diameter
Unbounded from distance to block
Unbounded from distance to co-cluster on the complement
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 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 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 cochromatic number
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

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

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 cochromatic 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 cochromatic number
Unbounded from cutwidth
Unbounded from degeneracy
Unbounded from distance to block
Unbounded from distance to linear forest
Unbounded from distance to outerplanar
Unbounded from maximum clique
Unbounded from maximum degree
Unbounded from pathwidth
Unbounded from 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 ∩ interval [trivial]
Unbounded on hamiltonian ∩ split [trivial]
Unbounded on square of tree

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 cochromatic number
Unbounded from degeneracy
Unbounded from maximum clique

Unbounded on (C₄,P₃,triangle)-free [by definition]
Unbounded on caterpillar [by definition]
Unbounded on complete [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 SC 2-tree [trivial]
Unbounded on SC 3-tree [trivial]
Unbounded on disjoint union of stars [by definition]
Unbounded on square of tree

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 cluster [trivial]
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 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 cochromatic number
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 clique
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

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

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

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 cochromatic number
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 cochromatic number
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 clique
Unbounded from maximum induced matching
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.

Polynomial

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

[119]

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

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

[1419]

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

Polynomial on weakly chordal

[1421]

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

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.

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.

Polynomial

Polynomial from Independent set on the complement
Polynomial from Weighted clique

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 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

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

[453]

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

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 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 [$O(V^2)$]

[1424]

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

Polynomial on circular perfect

[1408]

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 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 [$O(V^4E)$] 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.

NP-complete

[1146]

K.S. Booth, J.H. Johnson
Dominating sets in chordal graphs
SIAM J. Comput. 11 (1982) 191-199

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

NP-complete on undirected path

[1146]

K.S. Booth, J.H. Johnson
Dominating sets in chordal graphs
SIAM J. Comput. 11 (1982) 191-199

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

[1574]

P. Festa, P.M. Pardalos, M.G.C. Resende
Feedback set problems
in: D.Z. Du, P.M. Pardalos, Handbook of Combinatorial Optimization, Supplement vol. A, Kluwer Academic Publishers, 209-259 (2000)

Polynomial on chordal ∪ co-chordal

Graph isomorphism

[?]

Input: Graphs G and H in this class

Output: True iff G and H are isomorphic.

GI-complete

[1688]

V.N. Zemlyachenko, N.M. Korneenko, R.I. Tyshkevich
Graph isomorphism problem
J. of Soviet Mathematics 29 No.4 1426-1481

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

GI-complete on strongly chordal

[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

[1525]

C.J. Colbourn, L.K. Stewart
Dominating cycles in series-parallel graphs
Ars Combin. 19A 107-112 (1985)

NP-complete on directed path

[1526]

B.S. Panda, D. Pradhan
NP-Completeness of Hamiltonian Cycle Problem on Rooted Directed Path Graphs
Manuscript

NP-complete on rooted directed path

[1519]

G. Narasimhan
A note on the Hamiltonian Circuit Problem on directed path graphs
Information Proc. Lett. 32 No.4 167-170 (1989)

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)

NP-complete on undirected path

[1520]

A.A. Bertossi, M.A. Bonuccelli
Hamiltonian Circuits in interval graph generalizations
Information Proc. Lett. 23 195-200 (1986)

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 rooted directed path

[1519]

G. Narasimhan
A note on the Hamiltonian Circuit Problem on directed path graphs
Information Proc. Lett. 32 No.4 167-170 (1989)

NP-complete on split ∩ strongly chordal

[1517]

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

NP-complete on undirected path

[1520]

A.A. Bertossi, M.A. Bonuccelli
Hamiltonian Circuits in interval graph generalizations
Information Proc. Lett. 23 195-200 (1986)

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

[1669]

M. Farber
Independent domination in chordal graphs
Operations Research Lett. 1 No.4 134-138 (1982)

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

[425]

F. Gavril
The intersection graphs of subtrees in trees are exactly the chordal graphs
J. Comb. Theory (B) 16 1974 47--56

[931]

D.J. Rose, R.E. Tarjan, G.S. Lueker
Algorithmic aspects of vertex elimination on graph
SIAM J. Computing 5 1976 266--283

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 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 [$O(VE)$] 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.

NP-complete

[1629]

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

NP-complete on interval

[1815]

R. Adhikary, K. Bose, S. Mukherjee, B. Roy
Complexity of maximum cut on interval graphs
Proc. of 37th International Symposium on Computation Geometry 7:1-7:11 (2021)

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)

NP-complete on undirected path

[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

[1767]

T. Ekim, P. Hell, J. Stacho, D. de Werra
Polarity of chordal graphs
Discrete Applied Mathematics 156 No. 13 2469-2479 (2008)

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)

Polarity

[?]

Input: A graph G in this class.

Output: True iff G is polar.

Polynomial

[1767]

T. Ekim, P. Hell, J. Stacho, D. de Werra
Polarity of chordal graphs
Discrete Applied Mathematics 156 No. 13 2469-2479 (2008)

Polynomial on chordal ∪ co-chordal

Recognition

[?]

Input: A graph G.

Output: True iff G is in this graph class.

Linear

[1020]

[931]

D.J. Rose, R.E. Tarjan, G.S. Lueker
Algorithmic aspects of vertex elimination on graph
SIAM J. Computing 5 1976 266--283

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)

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.

Linear

[1166]

A. Frank
Some polynomial algorithms for certain graphs and hypergraphs.
Proc. 5th Br. comb. Conf., Aberdeen 1975, Congr. Numer. XV, 211-226 (1976).

Polynomial on (K_2,3,P,hole)-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 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 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 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 [$O(V^4)$] on weakly chordal

[997]

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

Weighted maximum cut

[?]

(decision variant)

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

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

NP-complete

NP-complete 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)

NP-complete on P₃-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: chordal

References

Equivalent classes

Complement classes

Related classes

Inclusions

Map

Minimal superclasses

Maximal subclasses

Speed

Parameters

Problems

Parameter decomposition

Unweighted problems

Weighted problems