Connectivity for some families of composition networks

Connectivity of a connected graph G, κ(G), is an important index in exploring network topology which is the minimal number of vertices that need to be removed to separate G into disconnected or trivial. Let G₀,G₁,…,G_m−1 be m connected graphs of the same order. A matching composition network G is constructed by adding an arbitrary perfect matching between G₀ and G₁. For m≥3, a cycle composition network H is constructed by adding an arbitrary perfect matching between G_i and G_i+1(modm) for each 0≤i≤m−1. This construction has been so widely used in literature to build networks in which fault diagnosability can be studied, that it is worth to study their connectivity in detail, this is the main purpose of this paper. In this paper, we determine (1) κ(G)=δ(G) if κ(G₀)+κ(G₁)≥δ(G); otherwise, κ(G)≥κ(G₀)+κ(G₁), and (2) κ(H)=δ(H) if ∑_i=0^m−1κ(G_i)≥δ(H); otherwise, κ(H)≥∑_i=0^m−1κ(G_i). Examples show those bounds are tight. We then generalize these examples to a general composition using matchings on which we propose a conjecture on the connectivity and prove it for an important particular case.