Graphclass: neighbourhood-Helly

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.

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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 $\min_{i \colon V \rightarrow \mathbb{N}\;}\{\max_{\{u,v\}\in E\;} \{|i(u)-i(v)|\}\mid i\text{ is injective}\}$.

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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 $G$.

F. Bernhart, P.C. Kainen
The book thickness of a graph
J. of Combin. Th. (B) 27 320-331 (1979)

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.

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 $G$.

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.

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.

P. Erdos
Graph theory and probability
Canadian J. of Math. 11 34-38 (1959)

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 $j$.

R. Boliac, V.V. Lozin
On the clique-width of graphs in hereditary classes.
Rutcor Research Report RRR 14-2002

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

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

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

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)

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 $\overline{G}$.

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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 $\{v_{i+1}, \ldots, v_n\}$.

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.

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

The distance to block of a graph $G$ is the size of a smallest vertex subset whose deletion makes $G$ a block graph.

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.

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.

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.

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 .

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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.

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.

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.

The max-leaf number of a graph $G$ is the maximum number of leaves in a spanning tree of $G$.

The parameter maximum clique of a graph $G$ is the largest number of vertices in a complete subgraph of $G$.

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

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 $G$.

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 $G$.

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 $G$.

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.

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 $G$.

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 $G$.

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 $G$.

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.

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.

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.

Input:	A graph G in this class and an integer k.
Output:	True iff the book thickness of G is at most k.

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.

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.

Input:	A graph G in this class and an integer k.
Output:	True iff the cutwidth of G is at most k.

Input:	A graph G in this class and an integer k.
Output:	True iff the treewidth of G is at most k.

A. Brandst\"adt, V.D. Chepoi, F.F. Dragan
The algorithmic use of hypertree structure and maximum neighbourhood orderings
Discrete Appl. Math. 82 43--77 1998

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.

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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.

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Input:	A graph G in this class and an integer k.
Output:	True iff the vertices of G can be partitioned into k sets Si, such that whenever two vertices are in the same set Si, they are adjacent.

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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.

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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.

I.E. Zverovich, O.I. Zverovich
Independent Domination in hereditary classes
Theoretical Comp. Sci. 352 215-225 (2006)

I.E. Zverovich, V.E. Zverovich
An induced subgraph characterization of domination perfect graphs
J. Graph Theory 20 1995 375--395

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.

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Input:	Graphs G and H in this class
Output:	True iff G and H are isomorphic.

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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

Input:	A graph G in this class.
Output:	True iff G has a simple cycle that goes through every vertex of the graph.

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

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

Input:	A graph G in this class.
Output:	True iff G has a simple path that goes through every vertex of the graph.

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

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

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.

I.E. Zverovich, O.I. Zverovich
Independent Domination in hereditary classes
Theoretical Comp. Sci. 352 215-225 (2006)

I.E. Zverovich, V.E. Zverovich
An induced subgraph characterization of domination perfect graphs
J. Graph Theory 20 1995 375--395

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.

A. Brandst\"adt, V.D. Chepoi, F.F. Dragan
The algorithmic use of hypertree structure and maximum neighbourhood orderings
Discrete Appl. Math. 82 43--77 1998

(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.

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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)

Input:	A graph G in this class.
Output:	True iff G is monopolar.

V.B. Le, R. Nevries
Complexity and algorithms for recognizing polar and monopolar graphs
Theoretical Computer Science 528 1-11 (2014)

Input:	A graph G in this class.
Output:	True iff G is polar.

V.B. Le, R. Nevries
Complexity and algorithms for recognizing polar and monopolar graphs
Theoretical Computer Science 528 1-11 (2014)

Input:	A graph G.
Output:	True iff G is in this graph class.

F.F. Dragan
Centers of graphs and the Helly property
Dissertation Thesis, Moldova State University Chi\c sin\u au 1989

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.

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.

Input:	A graph G in this class with weight function on the vertices and a real k.
Output:	True iff G contains a set S of pairwise non-adjacent vertices, with the sum of the weights of the vertices in S at most k, such that every vertex in G is either in S or adjacent to a vertex in S.

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.

(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.

M. Kaminski
Max-cut and containment relations in graphs
Proceedings of WG 2010, Lecture Notes in Computer Science 6410, 15-26 (2010)

M. Kaminski
Max-cut and containment relations in graphs
Proceedings of WG 2010, Lecture Notes in Computer Science 6410, 15-26 (2010)