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  • Author: Gülnaz Boruzanli Ekinci x
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Abstract

A vertex cut of a connected graph G is a set of vertices whose deletion disconnects G. A connected graph G is super-connected if the deletion of every minimum vertex cut of G isolates a vertex. The super-connectivity is the size of the smallest vertex cut of G such that each resultant component does not have an isolated vertex. The Kneser graph KG(n, k) is the graph whose vertices are the k-subsets of {1, 2, . . . , n} and two vertices are adjacent if the k-subsets are disjoint. We use Baranyai’s Theorem on the decompositions of complete hypergraphs to show that the Kneser graph KG are super-connected when n ≥ 5 and that their super-connectivity is n ( n/2) − 6 when n ≥ 6.

Abstract

The domination gap of a graph G is defined as the di erence between the maximum and minimum cardinalities of a minimal dominating set in G. The term well-dominated graphs referring to the graphs with domination gap zero, was first introduced by Finbow et al. [Well-dominated graphs: A collection of well-covered ones, Ars Combin. 25 (1988) 5–10]. In this paper, we focus on the graphs with domination gap one which we term almost well-dominated graphs. While the results by Finbow et al. have implications for almost well-dominated graphs with girth at least 8, we extend these results to (C 3, C 4, C 5, C 7)-free almost well-dominated graphs by giving a complete structural characterization for such graphs.