Fault tolerance of hypercubes and folded hypercubes

Abstract

Let $$G = (V,E)$$ G = ( V , E ) be a connected graph. The conditional edge connectivity $$\lambda _\delta ^k(G)$$ ? ? k ( G ) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects $$G$$ G and each component of $$G - F$$ G - F has $$\delta \ge k$$ ? ? k . We assume that $$F \subseteq E$$ F ⊆ E is an edge set, $$F$$ F is called edge extra-cut, if $$G - F$$ G - F is not connected and each component of $$G - F$$ G - F has more than $$k$$ k vertices. The edge extraconnectivity $$\lambda _\mathrm{e}^k(G)$$ ? e k ( G ) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.