The super-connectivity of odd graphs and of their kronecker double cover

Abstract

The study of connectivity parameters forms an integral part of the research conducted in establishing the fault tolerance of networks. A number of variations on the classical notion of connectivity have been proposed and studied. In particular, the super--connectivity asks for the minimum number of vertices that need to be deleted from a graph in order to disconnect the graph without creating isolated vertices. In this work, we determine this value for two closely related families of graphs which are considered as good models for networks, namely the odd graphs and their Kronecker double cover. The odd graphs are constructed by taking all possible subsets of size $k$ from the set of integers $\{1,\ldots,2k+1\}$ as vertices, and defining two vertices to be adjacent if the corresponding $k$-subsets are disjoint; these correspond to the Kneser graphs $KG(2k+1,k)$. The Kronecker double cover of a graph $G$ is formed by taking the Kronecker product of $G$ with the complete graph on two vertices; in the case when $G$ is $KG(2k+1,k)$, the Kronecker double cover is the bipartite Kneser graph $H(2k+1,k)$. We show that in both instances, the super--connectivity is equal to $2k$.

Super-Connectivity of the Folded Locally Twisted Cube

“…The super-connectivity of G (or, respectively, the super edge connectivity), denoted by κ (1) (G) (or λ (1) (G)), is the minimum cardinality of all super vertex cuts (or super edge cuts) in G, if any exist. Many relevant results have been obtained regarding super-connectivity and super edge connectivity [3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”