Fractional Matching Preclusion Number Research Articles

Fractional matching preclusion numbers of Cartesian product graphs

The Cartesian product of two simple graphs G and H is the graph G□H whose vertex set is V(G)×V(H) and whose edge set is the set of all pairs (u1,v1)(u2,v2) such that either u1u2∈E(G) and v1=v2, or v1v2∈E(H) and u1=u2. The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph with no fractional perfect matching. In this paper, we determine fmp(G□H) when H is a cycle or a path of even order; Moreover, given any integers a,b with a≥1 and 0≤b≤a+1, we construct a graph G such that δ(G)=a and fmp(G□H)=b when H is a path of odd order.

Conditional fractional matching preclusion for burnt pancake graphs and pancake-like graphs

ABSTRACT The conditional fractional strong matching preclusion number of a graph G is the minimum size of F such that and G−F has neither a fractional perfect matching nor an isolated vertex. In this paper, we obtain the conditional fractional strong matching preclusion number for burnt pancake graphs and a subset of the class of pancake-like graphs.

Fractional matching preclusion number of graphs

The fractional matching preclusion number of a graph G, denoted by fmp(G), is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings. In this paper, we first give some sharp upper and lower bounds of fractional matching preclusion number. Next, graphs with large and small fractional matching preclusion number are characterized, respectively. In the end, we investigate some extremal problems on fractional matching preclusion number.

Fractional strong matching preclusion of some Cartesian product graphs

The fractional strong matching preclusion number of a graph is the minimum number of edges and vertices whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the fractional strong matching preclusion number for the Cartesian product of a graph and a cycle. As an application, the fractional strong matching preclusion number for torus networks is also obtained.

The fractional (strong) matching preclusion number of complete k-partite graph

The fractional (strong) matching preclusion number of a graph G, denoted by f(s)mp(G), is the minimum number of edges (and vertices) whose deletion results in a graph with no fractional perfect matching. Let Gn1,n2,…,nk be the complete k-partite graph, whose vertex set can be partitioned into k parts, each has ni(1≤i≤k) vertices and whose edge set contains all edges between two distinct parts. In this paper, we determine fmp(Gn1,n2,…,nk) and fsmp(Gn1,n2,…,nk).

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.

Conditional fractional matching preclusion of [formula omitted]-dimensional torus networks

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in the remaining graph that has no fractional perfect matchings. For many networks, their optimal fractional matching preclusion sets are precisely those edges incident with a single vertex. The probability that all failures concentrate around a vertex is often small. To overcome the shortcoming, we consider the concept of conditional fractional matching preclusion, in which isolated vertices are not permitted in fault networks. We establish the conditional fractional matching preclusion numbers and all possible minimum conditional fractional matching preclusion sets for n-dimensional torus networks with n≥3.

Fractional matching preclusion of product networks

The matching preclusion number of a graph, introduced in [2] as a fault analysis, is the minimum number of edges whose deletion leaves a resulting graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu [14] recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number of graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matching. If the sets of edges of the graph attaining the minimum are precisely those incident to a single vertex of minimum degree, we say such graph is fractional super matched. In this paper, the upper and lower bounds for the fractional matching preclusion number for Cartesian product, direct product, strong product, and lexicographic product are obtained, and we give sufficient conditions for such graphs to be fractional super matched.

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.

Fractional matching preclusion number of graphs and the perfect matching polytope

Let G be a graph with an even number of vertices. The matching preclusion number of G, denoted by mp(G), is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching. We introduced a 0–1 linear program which can be used to find the matching preclusion number of graphs. In this paper, by relaxing of the 0–1 linear program we obtain a linear program and call its optimal objective value as fractional matching preclusion number of graph G, denoted by mp f (G). We show mp f (G) can be computed in polynomial time for any graph G. By using the perfect matching polytope, we transform it into a new linear program whose optimal value equals the reciprocal of mp f (G). For bipartite graph G, we obtain an explicit formula for mp f (G) and show that ⌊ mp f (G) ⌋ is the maximum integer k such that G has a k-factor. Moreover, for any two bipartite graphs G and H, we show mp f (Gs H) ⩾ mp f (G) + ⌊ mp f (H) ⌋ , where Gs H is the Cartesian product of G and H. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.

Fractional Matching Preclusion for Möbius Cubes

Let F be an edge subset and F′ a subset of vertices and edges of a graph G. If G − F and G − F′ have no fractional perfect matchings, 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 size of FMP (FSMP) sets of G. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for the Möbius cube MQn. In adddition, all the optimal fractional strong preclusion sets of these graphs are categorized.

Fractional Matching Preclusion for Restricted Hypercube-Like Graphs

The restricted hypercube-like graphs, variants of the hypercube, were proposed as desired interconnection networks of parallel systems. The matching preclusion number of a graph is the minimum number of edges whose deletion results in the graph with neither perfect matchings nor almost perfect matchings. The fractional perfect matching preclusion and fractional strong perfect matching preclusion are generalizations of the matching preclusion. In this paper, we obtain fractional matching preclusion number and fractional strong matching preclusion number of restricted hypercube-like graphs, which extend some known results.

A Concise Survey of Matching Preclusion in Interconnection Networks

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. There are other related parameters and generalization including the strong matching preclusion number, the conditional matching preclusion number, the fractional matching preclusion number, and so on. In this survey, we give an introduction on the general topic of matching preclusion.

Fractional matching preclusion for arrangement graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu (2017) recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of G is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the FMP number and the FSMP number for arrangement graphs. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

Fractional matching preclusion for radix triangular mesh

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu recently introduced the concept of fractional matching preclusion number. The fractional matching preclusion number (FMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The fractional strong matching preclusion number (FSMP number) of [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices and edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we study the fractional matching preclusion number and the fractional strong matching preclusion number for the radix triangular mesh [Formula: see text], and all the optimal fractional matching preclusion sets and fractional strong matching preclusion sets of these graphs are categorized.

Fractional matching preclusion of the restricted HL-graphs

The fractional matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has no fractional perfect matchings, and the fractional strong matching preclusion number of a graph is the minimum number of edges and/or vertices whose deletion leaves a resulting graph with no fractional perfect matchings. In this paper, we determine these two numbers for the restricted HL-graphs.

Fractional Strong Matching Preclusion of Split-Star Networks

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.

Fractional Matching Preclusion for (n,k)-Star Graphs

The matching preclusion number of a graph is the minimum number of edges whose deletion results in a graph that has neither perfect matchings nor almost perfect matchings. As a generalization, Liu and Liu introduced the concept of fractional matching preclusion number in 2017. The Fractional Matching Preclusion Number (FMP number) of G is the minimum number of edges whose deletion leaves the resulting graph without a fractional perfect matching. The Fractional Strong Matching Preclusion Number (FSMP number) of G is the minimum number of vertices and/or edges whose deletion leaves the resulting graph without a fractional perfect matching. In this paper, we obtain the FMP number and the FSMP number for (n, k)-star graphs. In addition, all the optimal fractional strong matching preclusion sets of these graphs are categorized.

Fractional matching preclusion of graphs

Let F be an edge subset and \(F^{\prime }\) a subset of edges and vertices of a graph G. If \(G-F\) and \(G-F^{\prime }\) have no fractional perfect matchings, then F is a fractional matching preclusion (FMP) set and \(F^{\prime }\) is a fractional strong MP (FSMP) set of G. The FMP (FSMP) number of G is the minimum size of FMP (FSMP) sets of G. In this paper, the FMP number and the FSMP number of Petersen graph, complete graphs and twisted cubes are obtained, respectively.