Many-to-many two-disjoint path covers in restricted hypercube-like graphs

Abstract

A Disjoint Path Cover (DPC for short) of a graph is a set of pairwise (internally) disjoint paths that altogether cover every vertex of the graph. Given a set S of k sources and a set T of k sinks, a many-to-many k-DPC between S and T is a disjoint path cover each of whose paths joins a pair of source and sink. It is classified as paired if each source of S must be joined to a designated sink of T , or unpaired if there is no such constraint. In this paper, we show that every m-dimensional restricted hypercube-like graph with at most m − 3 faulty vertices and/or edges being removed has a paired (and unpaired) 2-DPC joining arbitrary two sources and two sinks where m ≥ 5. The bound m − 3 on the number of faults is optimal for both paired and unpaired types.

Fractional Matching Preclusion for Restricted Hypercube-Like Graphs

“…It has been attracted considerable attention due to its outstanding performance. For example, some embedding properties, especially Hamiltonian cycle and path embeddings of the restricted hypercube-like were studied in [8,9,11,17]. The matching preclusion number of the restricted hypercube-like graphs were determined in [16].…”