Fractional Matching Preclusion for Data Center Networks

An edge subset [Formula: see text] of [Formula: see text] is a fractional matching preclusion set (FMP set for short) if [Formula: see text] has no fractional perfect matchings. The fractional matching preclusion number (FMP number for short) of [Formula: see text], denoted by [Formula: see text], is the minimum size of FMP sets of [Formula: see text]. A set [Formula: see text] of edges and vertices of [Formula: see text] is a fractional strong matching preclusion set (FSMP set for short) if [Formula: see text] has no fractional perfect matchings. The fractional strong matching preclusion number (FSMP number for short) of [Formula: see text], denoted by [Formula: see text], is the minimum size of FSMP sets of [Formula: see text]. Data center networks have been proposed for data centers as a server-centric interconnection network structure, which can support millions of servers with high network capacity by only using commodity switches. In this paper, we obtain the FMP number and the FSMP number for data center networks [Formula: see text], and show that [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], [Formula: see text]. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.