Abstract
The matching preclusion number of graph G is the minimum size of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching. Let F be an edge subset and F′ be a subset of edges and vertices of a graph G. If G − F and G − F′ have no fractional matching preclusion, then F is a fractional matching preclusion (FMP) set, and F ′is a fractional strong matching preclusion (FSMP) set of G. The FMP (FSMP) number of G is the minimum number of FMP (FSMP) set of G. In this paper, we study fractional matching preclusion number and fractional strong matching preclusion number of split-star networks. Moreover, We categorize all the optimal fractional strong matching preclusion sets of split-star networks.
Highlights
We often write V(G) and E(G) are vertex set and edge set, respectively
The matching preclusion number of graph G is the minimum size of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching
The fractional matching preclusion (FMP) (FSMP) number of G is the minimum number of FMP (FSMP) set of G
Summary
We often write V(G) and E(G) are vertex set and edge set, respectively. Each edge of G is usually denoted by uv or vu. For graphs with an even number of vertices, an obvious matching preclusion set is the set of edges incident with a single vertex; such a set is called a trivial matching preclusion set. A set F of edges and vertices of G is a fractional strong matching preclusion set (FSMP set for short) if G − F has no fractional perfect matchings. The fractional strong matching preclusion number (FSMP number for short) of G, denoted by f smp(G), is the minimum size of FSMP sets of G, that is, f smp(G)=min{|F|: F is an FSMP set}. 2 n is introduced in (Cheng et al, 2001) which is the companion graph of An. In this paper, we study the fractional strong matching preclusion problem for the split-star graph. Because deletion of vertices is allowed, the analysis will be more involved than the analysis of the correspond matching preclusion problem
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