We prove that partitionable graphs are 2w -2-connected, that this bound is sharp, and prove some structural properties of cutsets of cardinality 2w -2. The proof of the connectivity result is a simple linear algebraic proof.
Some properties of minimal imperfect graphs
✍ Scribed by Chính T. Hoàng
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 539 KB
- Volume
- 160
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
📜 SIMILAR VOLUMES
IfI,: family of Bar, w) graphs ate of interest for several reasons. For example, any minimal fomenter-example to Rerge's Strong Perfect Graph Conjecture t %ngs to this family. This paper aciounts for ail (4.3) graphs. One of these is not obtainatde by existing techniques for geg~~rati~g (a + I, w) g
Results of Lovász and Padberg entail that the class of so-called partitionable graphs contains all the potential counterexamples to Berge's famous Strong Perfect Graph Conjecture, which asserts that the only minimal imperfect graphs are the odd chordless cycles with at least five vertices (''odd hol
V. Chva tal conjectured in 1985 that a minimal imperfect graph G cannot have a skew cutset (i.e., a cutset S decomposable into disjoint sets A and B joined by all possible edges). We prove here the conjecture in the particular case where at least one of A and B is a stable set. 2001 Elsevier Science
An edge of a graph is called critical, if deleting it the stability number of the graph increases, and a nonedge is called co-critical, if adding it to the graph the size of the maximum clique increases. We prove in this paper, that the minimal imperfect graphs containing certain configurations of t
We show that a minimal imperfect graph \(G\) cannot contain a cutset \(C\) which induces a complete multi-partite graph unless \(C\) is a stable set and \(G\) is an odd hole. This generalizes a result of Tucker, who proved that the only minimal imperfect graphs containing stable cutsets are the odd