The m-Component Connectivity of Leaf-Sort Graphs

Abstract

Connectivity plays an important role in measuring the fault tolerance of interconnection networks. As a special class of connectivity, m-component connectivity is a natural generalization of the traditional connectivity of graphs defined in terms of the minimum vertex cut. Moreover, it is a more advanced metric to assess the fault tolerance of a graph G. Let G=(V(G),E(G)) be a non-complete graph. A subset F(F⊆V(G)) is called an m-component cut of G, if G−F is disconnected and has at least m components (m≥2). The m-component connectivity of G, denoted by cκm(G), is the cardinality of the minimum m-component cut. Let CFn denote the n-dimensional leaf-sort graph. Since many structures do not exist in leaf-sort graphs, many of their properties have not been studied. In this paper, we show that cκ3(CFn)=3n−6 (n is odd) and cκ3(CFn)=3n−7 (n is even) for n≥3; cκ4(CFn)=9n−212 (n is odd) and cκ4(CFn)=9n−242 (n is even) for n≥4.