Two kinds of conditional connectivity of hypercubes

Abstract

A subset is called an h-extra r-component cut of G if G – F is disconnected and there are at least r components, each component has at least h + 1 vertices. The cardinality of a minimum h-extra r-component cut of G, denoted by is the h-extra r-component connectivity of G. In this paper, we introduce a novel connectivity called the g-good r-component connectivity. For if G – F is disconnected and there are at least r components and each vertex has at least g neighbors, then F is called a g-good r-component cut of G; the g-good r-component connectivity of G, denoted by is the minimum cardinality of a g-good r-component cut of G. In this work, we prove that for and for where Qn is n-dimension hypercube.