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
On minimal cutsets in P5-free minimal imperfect graphs
✍ Scribed by Vincent Barré; Jean-Luc Fouquet
- Book ID
- 108315605
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 115 KB
- Volume
- 236
- Category
- Article
- ISSN
- 0012-365X
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proved that no minimal imperfect graph has a small transversal, that is, a set of vertices of cardinality at most x + M-1 which meets every c+clique and every x-stable set. In this paper we prove that a slight generalization of this notion of small transversal leads to a conjecture which is as stro
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
## Abstract It is shown that the minimum number of vertices in a triangle‐free 5‐chromatic graph is at least 19.