G. A. Dirac characterized chordal graphs as those in which minimal vertex separators always induce complete subgraphs. I generalize a traditional (2-)vertex separator to a k-vertex separator — meaning a set S of vertices whose removal puts k independent vertices into k separate components. Generalizing Dirac’s theorem, the {P5, 2P3}-free chordal graphs are the graphs in which minimal k-separato...

We give three constructions of a vertex-minimal triangulation of 4-dimensional real projective space RP4. The first construction describes a 4-dimensional sphere on 32 vertices, which is a double cover of a triangulated RP4 and has a large amount of symmetry. The second and third constructions illustrate approaches to improving the known number of vertices needed to triangulate n-dimensional re...

Lubiw [11] conjectures that in a minimal imperfect Berge graph, the neighborhood graph N (v) of any vertex v must be connected; this conjecture implies a well known Chvátal’s Conjecture [6] which states that N (v) must contain a P4. In this note we will prove an intermediary conjecture for some classes of minimal imperfect graphs. It is well known that a graph is P4-free if, and only if, every ...

Let c (k) (c. (k» be the least number of vertices of a planar (planar connected) graph whose automorphism group is the cyclic group of order k. If k r is odd and if k = PIal. . . p,ar, PI' . . . ,Pr being primes then c (k) = 3 2: Piai and i=l c. (k) ~ c (k) for r ~ 1 while c. (k) = c (k) + 1 for r> 1. It is conjectured that a similar result holds for k even.

A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we characterize minimal cograph completions, and we give the following linear-time algorithms: one for extracting a minimal cograph completion from any given cograph completion of G, and one for d...

We study minimal vertex covers of trees. Contrarily to the number Nvc(A) of minimal vertex covers of the tree A, log Nvc(A) is a self-averaging quantity. We show that, for large sizes n, limn→+∞ < log Nvc(A) >n /n = 0.1033252 ± 10−7. The basic idea is, given a tree, to concentrate on its degenerate vertices, that is those vertices which belong to some minimal vertex cover but not to all of them...

A graph is vertex-critical (edge-critical) if deleting any vertex (edge) increases its diameter. A conjecture of Simon and Murty stated thatèvery edge-critical graph of diameter two on vertices contains at most 1 4 2 edges'. This conjecture has been established for suuciently large. For vertex-critical graphs, little is known about the number of edges. Plesn ik implicitly asked whether it is al...

This paper introduces the concept of Vertex Unique Labelled Subgraph Mining (VULSM), a specialised form of subgraph mining. A VULS is a subgraph defined by a set of edge labels that has a unique vertex labelling associated with it. A minimal VULS is then a VULS which is not a supergraph of any other VULS. The application considered in this paper, for evaluation purposes, is error prediction wit...

We show that minimal k-vertex connected spanning subgraphs of a given graph can be generated in incremental polynomial time for any fixed k.