# Search Results

## 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.