The Component Connectivity of Alternating Group Graphs and Split-Stars
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For an integer ℓ≥ 2, the ℓ-component connectivity of a graph G, denoted by κ_ℓ(G), is the minimum number of vertices whose removal from G results in a disconnected graph with at least ℓ components or a graph with fewer than ℓ 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 ℓ-connectivity are known only for a few classes of networks and small ℓ's. It has been pointed out in [Component connectivity of the hypercubes, Int. J. Comput. Math. 89 (2012) 137--145] that determining ℓ-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_n^2, we study their ℓ-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_n^2)=4n-8, κ_4(S_n^2)=6n-14, and κ_5(S_n^2)=8n-20 for n≥ 4.
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