This paper studies the concepts of uniquely colorable graphs & Perfect graphs. The main results are 1) Every uniquely k-colorable graph is (k 1)-connected. 2) If G is a uniquely k-colorable graph, then  (G) ≥ k l. 3) A maximal planar graph G of order 3 or more has chromatic number 3 if and only if G is Eulerian. 4) Every interval graph is perfect. 5) A graph G is chordal if and only if G can b...

Applied to a chordal graph, lexicographic breadth rst search computes a perfect elimination scheme. We present a short elegant proof for this fact using only a new property of separators in chordal graphs.

Maxcliques (maximal complete subgraphs) and unit disks (closed neighborhoods of vertices) sometime play almost interchangeable roles in graph theory. For instance, interchanging them makes two existing characterizations of chordal graphs into two new characterizations. More intriguingly, these characterizations of chordal graphs can be naturally strengthened to new characterizations of strongly...

Elias Dahlhaus Paul D. Manuel y Mirka Miller y Abstract In this paper, we present a simple charactrization of strongly chordal graphs. A chordal graph is strongly chordal if and only if every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle.

Many works related to dually chordal graphs, their cliques and neighborhoods were published [1, 6]. We will undertake a similar study by considering minimal vertex separators and their properties instead. We find a necessary and sufficient condition for every minimal vertex separator to be contained in the closed neighborhood of a vertex and two major characterizations of dually chordal graphs ...

A graph G is dually chordal if there is a spanning tree T of G such that any maximal clique of G induces a subtree in T . This paper investigates the Colourability problem on dually chordal graphs. It will show that it is NP-complete in case of four colours and solvable in linear time with a simple algorithm in case of three colours. In addition, it will be shown that a dually chordal graph is ...

Let k be an integer and k ≥ 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G is chordal then so is G. Brandstädt et al. in [Andreas Brandstädt, Van Bang Le, and Thomas Szymczak. Duchet-type theorems for pow...

We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. The main contribution is a construction which shows that all chordal graphs which have a representation as intersection graph of subpaths on a tree are representable. A family of intersection graphs of substars of a star is used to show that not all chordal graphs are representable by pseudosegme...

The minimal interval completion problem consists in adding edges to an arbitrary graph so that the resulting graph is an interval graph; the objective is to add an inclusion minimal set of edges, which means that no proper subset of the added edges can result in an interval graph when added to the original graph. We give an O(n2)-time algorithm to obtain a minimal interval completion of an arbi...

We describe a greedy vertex colouring method which can be used to colour optimally the edge set of certain chordal graphs. This new heuristic yields an exact edge-colouring algorithm for odd maximumdegree doubly chordal graphs. This method shows that any such graph G can be edge-coloured with maximum degree (G) colours, i.e., all these graphs are Class 1. In addition, this method gives a simple...