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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(n, 2) are super-connected when n ≥ 5 and that their super-connectivity is n 2 − 6 when n ≥ 6.

The 2-domination number γ 2 (G) of a graph G is the minimum cardinality of a set D ⊆ V (G) for which every vertex outside D is adjacent to at least two vertices in D. Clearly, γ 2 (G) cannot be smaller than the domination number γ(G). We consider a large class of graphs and characterize those members which satisfy γ 2 = γ. For the general case, we prove that it is NP-hard to decide whether γ 2 = γ holds. We also give a necessary and sufficient condition for a graph to satisfy the equality hereditarily.

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Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.