A cycle C in a graph G is extendible if there exists a cycle C 0 in G such that V (C) V (C 0) and jV (C 0)j = jV (C)j + 1. A graph G is cycle extendible if G contains at least one cycle and every nonhamiltonian cycle in G is extendible. A graph G is fully cycle extendible if G is cycle extendible and every vertex lies on a 3-cycle of G. G. Hendry asked if every hamiltonian chordal graph is full...

In a graph G, the p-center problem is to identify p vertices of G for locating some kind of facilities such that the maximum among the distances from all vertices to their nearest facility is minimized. In this paper, we propose an O(n)-time algorithm for computing a 2-center in a chordal graph with maximum degree at most three, where a graph is chordal if every induced cycle of length at least...

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft [13] introduced connected matchings and proved that, given a positive integer k, determining whether a graph has a connected matching of size at least k is NPcomplete. Cameron [4] proved that ...

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs. One of them is the class of hereditary basic chordal graphs, which will turn out to have many p...

A graph is H-free if it has no induced subgraph isomorphic to H . Brandstädt, Engelfriet, Le and Lozin proved that the class of chordal graphs with independence number at most 3 has unbounded clique-width. Brandstädt, Le and Mosca erroneously claimed that the gem and the co-gem are the only two 1-vertex P4-extensions H for which the class of H-free chordal graphs has bounded clique-width. In fa...

Let G be a graph with the usual shortest-path metric. A graph is δ-hyperbolic if for every geodesic triangle T , any side of T is contained in a δ-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. In this paper we study the relation between the hyperbolicity of the graph and some chordality properties which are natural generaliz...

Almost self-centered graphs were recently introduced as the graphs with exactly two non-central vertices. In this paper we characterize almost selfcentered graphs among median graphs and among chordal graphs. In the first case P4 and the graphs obtained from hypercubes by attaching to them a single leaf are the only such graphs. Among chordal graph the variety of almost self-centered graph is m...

A graph class is sandwich monotone if, for every pair of its graphs G1 = (V,E1) and G2 = (V,E2) with E1 ⊂ E2, there is an ordering e1, . . . , ek of the edges in E2 \E1 such that G = (V,E1 ∪ {e1, . . . , ei}) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakon...

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chord...

The class Lk of k-leaf powers consists of graphs G = (V,E) that have a k-leaf root, that is, a tree T with leaf set V , where xy ∈ E, if and only if the T -distance between x and y is at most k. Structure and linear time recognition algorithms have been found for 2-, 3-, 4-, and, to some extent, 5-leaf powers, and it is known that the union of all k-leaf powers, that is, the graph class L = ⋃∞ ...