# Graphclass: house-free ∩ weakly chordal

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## 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.

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## 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 [+]Details 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 [+]Details 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 [+]Details 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 [+]Details branchwidth [?] A branch decomposition of a graph $G$ is a pair $(T,\chi)$, 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 $X, E \backslash X$, 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 [+]Details 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 [+]Details 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 [+]Details 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 $i$, disjoint union, renaming labels $i$ to label $j$, and connecting all vertices with label $i$ to all vertices with label $j$. Unbounded [+]Details 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 $i = 1, \ldots,n - 1$, 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 [+]Details degeneracy [?] The degeneracy of a graph $G$ is the smallest integer $k$ such that each subgraph of $G$ contains a vertex of degree at most $k$. Unbounded [+]Details diameter [?] The diameter of a graph $G$ is the length of the longest shortest path between any two vertices in $G$. Unbounded [+]Details 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 [+]Details 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 [+]Details 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 [+]Details 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 [+]Details 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 [+]Details 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 [+]Details distance to outerplanar [?] The distance to outerplanar of a graph $G = (V,E)$ is the minumum size of a vertex subset $X \subseteq V$, such that $G[V \backslash X]$ is a outerplanar graph. Unbounded [+]Details 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 [+]Details maximum clique [?] The parameter maximum clique of a graph $G$ is the largest number of vertices in a complete subgraph of $G$. Unbounded [+]Details maximum degree [?] The maximum degree of a graph $G$ is the largest number of neighbors of a vertex in $G$. Unbounded [+]Details 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 [+]Details 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 [+]Details 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 [+]Details max-leaf number [?] The max-leaf number of a graph $G$ is the maximum number of leaves in a spanning tree of $G$. Unbounded [+]Details 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 [+]Details 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 [+]Details pathwidth [?] A path decomposition of a graph $G$ is a pair $(P,X)$ where $P$ is a path with vertex set $\{1, \ldots, q\}$, 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 [+]Details 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 [+]Details tree depth [?] A tree depth decomposition of a graph $G = (V,E)$ is a rooted tree $T$ with the same vertices $V$, such that, for every edge $\{u,v\} \in E$, 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 [+]Details treewidth [?] A tree decomposition of a graph $G$ is a pair $(T, X)$, 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 $X_i$, $i \in I$ equals $V$, for all edges $\{v,w\} \in E$, there exists $i \in I$, such that $v, w \in X_i$, 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 [+]Details 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 [+]Details

## 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 Unknown to ISGCI [+]Details
booleanwidth decomposition Unknown to ISGCI [+]Details
cliquewidth decomposition Unknown to ISGCI [+]Details
cutwidth decomposition NP-complete [+]Details
treewidth decomposition Polynomial [+]Details

#### Unweighted problems

3-Colourability Polynomial [+]Details
Clique Polynomial [+]Details
Clique cover Polynomial [+]Details
Colourability Polynomial [+]Details
Domination NP-complete [+]Details
Feedback vertex set Unknown to ISGCI [+]Details
Graph isomorphism GI-complete [+]Details
Hamiltonian cycle NP-complete [+]Details
Hamiltonian path NP-complete [+]Details
Independent set Polynomial [+]Details
Maximum cut NP-complete [+]Details
Monopolarity Polynomial [+]Details
Polarity Unknown to ISGCI [+]Details
Recognition Polynomial [+]Details

#### Weighted problems

Weighted clique Polynomial [+]Details
Weighted feedback vertex set Unknown to ISGCI [+]Details
Weighted independent set Polynomial [+]Details
Weighted maximum cut NP-complete [+]Details