The Component Connectivity of Alternating Group Graphs and Split-Stars

Abstract

For an integer 2, the -component connectivity of a graph G, denoted by κ (G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least components or a graph with fewer than vertices. This is a natural generalization of the classical connectivity of graphs defined in term of the minimum vertex-cut and is a good measure of robustness for the graph corresponding to a network. So far, the exact values of -connectivity are known only for a few classes of networks and small 's. It has been pointed out in [Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137-145] that determining -connectivity is still unsolved for most interconnection networks, such as alternating group graphs and star graphs. In this paper, by exploring the combinatorial properties and fault-tolerance of the alternating group graphs AG n and a variation of the star graphs called split-stars S 2 n , we study their -component connectivities. We obtain the following results: (i) κ 3 (AG n ) = 4n−10 and κ 4 (AG n ) = 6n−16 for n 4, and κ 5 (AG n ) = 8n − 24 for n 5; (ii) κ 3 (S 2 n ) = 4n − 8, κ 4 (S 2 n ) = 6n − 14, and κ 5 (S 2 n ) = 8n − 20 for n 4.