On super connectivity of Cartesian product graphs

Abstract

The super connectivity κ 1 of a connected graph G is the minimum number of vertices whose deletion results in a disconnected graph without isolated vertices; this is a more refined index than the connectivity parameter κ. This article provides bounds for the super connectivity κ 1 of the Cartesian product of two connected graphs, and thus generalizes the main result of Shieh on the super connectedness of the Cartesian product of two regular graphs with maximum connectivity. Particularly, we determine that κ 1 (K m × K n ) = min{m + 2n − 4, 2m + n − 4} for m + n ≥ 6 and state sufficient conditions to guarantee κ 1 (K 2 × G) = 2κ(G). As a consequence, we immediately obtain the super connectivity of the n-cube for n ≥ 3.

Super-connected but not super edge-connected graphs

“…For more results on super-connected graphs or super edge-connected graphs, we refer the reader to [6,7,10,11,13,[15][16][17].…”

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

“…In this case, κ is strictly greater than the connectivity κ = κ(G) of G; otherwise κ = κ. Some examples of graph classes which have been analysed for their super-connectivity are circulant graphs [3], products of various graphs (see [7,8,12,19], and the references therein), hypercubes [13,24,25], generalized Petersen graphs [4], Johnson graphs [5] and Kneser graphs [1,6].…”

The Super-Connectivity of the Double Vertex Graph of Complete Bipartite Graphs

“…If such a vertex-cut exists, it is referred to as a super vertex-cut; otherwise we write κ (G) = +∞. The super-connectivity has been studied for various families of graphs, including circulant graphs [16], hypercubes [17,18], product graphs [19]- [21].…”