Maximally m-induced Subgraph of Some Interconnection Networks

Abstract

The topological structure of an interconnection network can be modeled by a graph G = (V;E) where V is the vertex set and E the edge set of G. For a vertex subset V 0 µ V of graph G, the subgraph of G induced by V 0, denoted by G[V 0], is a graph with vertex set V 0 and all the edges of G with both ends of vertices in V 0. An m- induced subgraph of a graph is such one which induced by m vertices. A maximally m-induced subgraph of a graph G, denoted by V max m (G), can be defined as V max m (G) = fG[V 0] j maxV 0µV;jV 0j=m jE(G[V 0])jg. Let maxm(G) be the number of edges in such a maximally m-induced subgraph V max m (G). Let m be an integer with m = Pr¡1 i=0 2li and l0 > l1 > ¢ ¢ ¢ > lr¡1. g(m) = Pr¡1 i=0 ( li 2 + i)2li . For an n-dimensional hypercube Qn, it is proved by Abdel- Ghaffar in 2003 that maxm(Qn) = g(m) for n ¸ 1 and 0 · m · 2n. In this paper, we investigate in the maximally m-induced subgraph of the generalized hypercubes GQn and show that maxm(GQn) = g(m) for n ¸ 3 and 0 · m · 2n. The hypercubes, twisted cubes, crossed cubes, and m¨obius cubes are special cases of generalized hypercubes. Additionally, we provide an algorithm to find a maximally m-induced subgraph of generalized hypercubes.