We show that if X is a chordal graph containing no clique of size μ (μ an infinite cardinal) then the chromatic (even coloring) number of X is at most μ. The same conclusion holds if the condition ‘is chordal’ is replaced by ‘contains no induced C4 (or Kk,k for k finite)’. In [9] Wagon asked if the following holds. If X is a chordal graph with Kω 6≤ X then Chr(X) ≤ ω. This was proved by Halin i...

Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, the so-called tree model of G. The leafage l(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G. This concept was first defined by I.-J. Lin, T.A. McKee, and D.B. West in [9]. In this contribution, we present the first polynomial tim...

A blocking quadruple (BQ) is a quadruple of vertices of a graph such that any two vertices of the quadruple either miss (have no neighbours on) some path connecting the remaining two vertices of the quadruple, or are connected by some path missed by the remaining two vertices. This is akin to the notion of asteroidal triple used in the classical characterization of interval graphs by Lekkerkerk...

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of ind...

For a chordal graph G = (V, E), we study the problem of whether a new vertex u 6∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of ex...

We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G−B), where B are the bridges of G. In particular, it follows that for strongly chordal graphs and so-called power chordal graphs the injective chromatic number can be determined in polynomial time. Moreover, for chordal graphs in general, we show ...

in this paper, we introduce a subclass of chordal graphs which contains $d$-trees and show that their complement are vertex decomposable and so is shellable and sequentially cohen-macaulay.

Thl.: use of (generalized) tree structure in graphs is one of the main topics in the field of eflicient graph algorithms. The well-known partial k-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are k...

A chordal graph is the intersection graph of a family of subtrees of a tree, or, equivalently, it is the (edge-)intersection graph of leaf-generated subtrees of a full binary tree. In this paper, a generalization of chordal graphs from this viewpoint is studied: a graph G=(V; E) is representable if there is a family of subtrees {Sv}v∈V of a binary tree, such that uv ∈ E if and only if |Su ∩ Sv|...

A graph is chordal if every induced cycle has exactly three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. A graph is δ-hyperbolic if every geodesic triangle is δ-thin. In this paper, we study the relation between vertex separator sets, certain chordality properties that generalize being chordal and the hyperbolicity of the graph. We also give a cha...