Abstract
A graph G = (V, E) is F-strongly Menger-vertex-connected (resp. Menger-edge-connected), if for a subgraph G - F of G with a conditional faulty vertex (resp. edge) set F subset of V (resp.
F subset of E), each pair of vertices u and v are connected by min(deg(G-F) (u), deg(G-F )(v)} internally disjoint (resp. edge-disjoint) fault-free paths in G - F. Where deg(G-F) (u) and deg(G-F) (v) are the degrees of u and v in G - F, respectively.
He et al. (2018) [7] introduced some sufficient conditions of F-strongly Menger (edge) connected for k-regular triangle-free graphs. In this paper, we consider the strongly Menger connectedness of two kinds of graphs with triangles: data center network D-k,D-n and (n, k)-star graph S-n.k.
We prove D-k,D- n is (k - 1)-strongly Menger-vertex-connected, (n + k - 3)-strongly Mengeredge-connected of order 1, (2n + 2k - 8)-strongly Menger-edge-connected of order 2, and (3n + 3k - 15)-strongly Menger-edge-connected of order 3. Furthermore, we obtain S-n,S-k is (k - 2)-strongly Menger-vertex-connected, (n - 3)-strongly Menger-edge-connected of order 1, (2n - 8)-strongly Menger-edge-connected of order 2, and (3n - 15)-strongly Menger-edge-connected of order 3. (C) 2019 Elsevier B.V.
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