Conditional Matching Preclusion Number Research Articles

Conditional Fractional Matching Preclusion Number of Graphs

The conditional fractional matching preclusion number (CFMP number for short) [Formula: see text] of a graph [Formula: see text] is the minimum number of edges whose deletion results in a graph without isolated vertices and without fractional perfect matchings. In this paper, we study the CFMP number of complete graphs, complete bipartite graphs and twisted cubes. Also, we give Nordhaus–Gaddum-type results for the CFMP numbers of general graphs.

Conditional Matching Preclusion Number of Graphs

The conditional matching preclusion number of a graph G , denoted by m p 1 G , is the minimum number of edges whose deletion results in the graph with no isolated vertices that has neither perfect matching nor almost-perfect matching. In this paper, we first give some sharp upper and lower bounds of conditional matching preclusion number. Next, the graphs with large and small conditional matching preclusion numbers are characterized, respectively. In the end, we investigate some extremal problems on conditional matching preclusion number.

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.

Conditional strong matching preclusion of the pancake graph

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this article, we find the conditional strong matching preclusion number for the pancake graph.

The Matching Preclusion of Enhanced Hypercubes

Abstract The (conditional) matching preclusion number of a graph is the minimum number of edges whose deletion leaves the resulting graph (with no isolated vertices) that has neither perfect matchings nor almost perfect matchings. The (conditional) strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion makes the resulting graph (with no isolated vertices) without perfect matching or almost perfect matching. The enhanced hypercube $Q_{n,k}$ $(1\leq k\leq n-1)$ is an extension of hypercube. In this paper, we prove that the matching preclusion number of $Q_{n,k}$ is $n+1$ $(1\leq k\leq n-1)$, the strong matching preclusion number of $Q_{n,k}$ is $n+1$ $(2\leq k\leq n-1)$, the conditional matching preclusion number of $Q_{n,n-1}$ is $2n-1$, the conditional matching preclusion number of $Q_{n,k}$ is $2n$ $(1\leq k\leq n-2)$ and the conditional strong matching preclusion number of $Q_{n,n-2}$ is $2n-3$ $(n\geq 4)$.

Conditional matching preclusion for regular bipartite graphs and their Cartesian product

Let G be a graph with an even number of vertices. The conditional matching preclusion number of G is the minimum number of edges whose deletion results in a graph with neither isolated vertices nor perfect matchings. Conditional matching preclusion number was introduced as an improvement of matching preclusion number for measuring the robustness of a network when there is a link failure with a higher accuracy. In this paper, we characterize all conditionally maximally matched regular bipartite graphs and conditionally super matched regular bipartite graphs, respectively. Furthermore, for r1-regular bipartite graph G and r2-regular bipartite graph H with r1⩾3 and r2⩾3, we show that if G is conditionally maximally matched and H is r2∕2-edge-connected, then G□H is conditionally super matched except for one class of graphs.

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.

Connectivity and Matching Preclusion for Leaf-Sort Graphs

A multiprocessor system and interconnection network have a underlying topology, which is usually presented by a graph, where nodes represent processors and links represent communication links between processors. The (conditional) matching preclusion number of a graph is the minimum number of edges whose deletion leaves the resulting graph (with no isolated vertices) that has neither perfect matchings nor almost perfect matchings. In this paper, we prove that (1) the connectivity (edge connectivity) of the leaf-sort graph CFn is [Formula: see text] for odd n and [Formula: see text] for even n; (2) CFn is super edge-connected; (3) the matching preclusion number of CFn is [Formula: see text] for odd n and [Formula: see text] for even n; (4) the conditional matching preclusion number of CFn is 3n – 5 for odd n and n ≥ 3, and 3n – 6 for even n and n ≥ 4.

Conditional Matching Preclusion for Folded Hypercubes

Let G be a graph with an even number of vertices. The matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion results in a graph with no isolated vertices and without a perfect matching. Matching preclusion number was introduced for measuring the robustness of a network when there is a link failure. In this paper, we focus on conditional matching preclusion for folded hypercube FQn, an important variant of hypercube. We show that conditional matching preclusion number of FQn is 2n and all optimal conditional matching preclusion sets are trivial for n ⩾ 5.

A note on generalized matching preclusion in bipartite graphs

For a graph with an even number of vertices, the matching preclusion number is the minimum number of edges whose deletion results in a graph with no perfect matchings. The conditional matching preclusion number, introduced as an extension of the matching preclusion number, has the additional requirement that the resulting graph have no isolated vertices. In bipartite graphs, the edge sets that achieve these minimum numbers can be construed in terms of obstruction sets of vertices. From this perspective, the generalized matching preclusion problem was developed; the matching preclusion number of order a is the minimum number of edges whose deletion results in a graph with no perfect matchings and no obstruction sets with fewer than a vertices. In this paper, we extend known sufficient conditions regarding the matching preclusion and conditional matching preclusion numbers (orders 1 and 2, respectively) to give sufficient conditions for the matching preclusion problem of order a, for any positive integer a.

Conditional Strong Matching Preclusion of the Alternating Group Graph

The strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings. Park and Ihm introduced the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. In this paper, we find the conditional strong matching preclusion number for the $n$-dimensional alternating group graph $AG_n$.

Matching Preclusion Number of Radix Triangular Mesh with Odd Vertices

The matching preclusion number of graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or almost-perfect matching. The strong matching preclusion number of a graph G is the minimum number of vertices and edges whose deletion leaves the resulting graph without a perfect matching or an almost-perfect matching. The conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without a perfect matching or almost-perfect matching. In this paper, we study the matching preclusion number of radix triangular mesh with an odd number of vertices, and strong matching preclusion number and conditional matching preclusion number of radix triangular mesh. Also, we obtained the radix triangular mesh with an even number of vertices is super strongly matched and conditionally super matched.

Matching preclusion and conditional edge-fault Hamiltonicity of binary de Bruijn graphs

In interconnection networks, matching preclusion is a measure of robustness when there is a link failure. Let G be a graph with an even number of vertices. The matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion results in a graph with no isolated vertices and without a perfect matching. In this paper, we study these problems for undirected binary de Bruijn graph UB(n). As an essential preparation, we obtain conditional edge-fault Hamiltonicity of binary de Bruijn graph B(n). Then we obtain matching preclusion number and conditional matching preclusion number of UB(n) for n⩾4 and classify all optimal matching preclusion sets and optimal conditional matching preclusion sets.

Matching preclusion for cube-connected cycles

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. The matching preclusion number of a graph G with even order is the minimum number of edges whose deletion results in a graph without perfect matchings and the conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resulting graph with no isolated vertices and without perfect matchings. We consider matching preclusion of cube-connected cycles network CCCn. By using the super-edge-connectivity of vertex-transitive graphs, the super cyclically edge-connectivity of CCCn for n=3,4 and 5, Hall’s Theorem and the strengthened Tutte’s Theorem, we obtain the matching preclusion number and the conditional matching preclusion number of CCCn and classify respective optimal matching preclusion sets.

Matching preclusion and conditional matching preclusion problems for the folded Petersen cube

The matching preclusion number of an even graph is the minimum number of edges whose deletion results in a graph that has no perfect matchings. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of an even graph was introduced to look for obstruction sets beyond those induced by a single vertex. It is defined to be the minimum number of edges whose deletion results in a graph with no isolated vertices and no perfect matchings. In this paper, we study this problem for the folded Petersen cube FPQ(n,k) via some matching preclusion and conditional matching preclusion results of the Cartesian products of graphs.

Matching preclusion for augmented k-ary n-cubes

The (conditional) matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph (with no isolated vertices) that has neither perfect matchings nor almost perfect matchings. In this paper, we find this number and classify all optimal sets for the augmented k -ary n-cubes with even k ⩾ 4.

Matching preclusion and conditional matching preclusion for pancake and burnt pancake graphs

The matching preclusion number of a graph with an even number of vertices is the minimum number of edges whose deletion destroys all perfect matchings in the graph. The optimal matching preclusion sets are often precisely those which are induced by a single vertex of minimum degree. To look for obstruction sets beyond these, the conditional matching preclusion number was introduced, which is defined similarly with the additional restriction that the resulting graph has no isolated vertices. In this paper we find the matching preclusion and conditional matching preclusion numbers and classify all optimal sets for the pancake graphs and burnt pancake graphs.

CONDITIONAL MATCHING PRECLUSION FOR (n,k)-STAR GRAPHS

The matching preclusion number of an even graph G, denoted by mp (G), is the minimum number of edges whose deletion leaves the resulting graph without perfect matchings. The conditional matching preclusion number of an even graph G, denoted by mp 1(G), is the minimum number of edges whose deletion leaves the resulting graph with neither perfect matchings nor isolated vertices. The class of (n,k)-star graphs is a popular class of interconnection networks for which the matching preclusion number and the classification of the corresponding optimal solutions were known. However, the conditional version of this problem was open. In this paper, we determine the conditional matching preclusion for (n,k)-star graphs as well as classify the corresponding optimal solutions via several new results. In addition, an alternate proof of the results on the matching preclusion problem will also be given.

The (conditional) matching preclusion for burnt pancake 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, and the conditional matching preclusion number of a graph is the minimum number of edges whose deletion leaves a resulting graph with no isolated vertices that has neither perfect matchings nor almost perfect matchings. In this paper, we find these two numbers for the burnt pancake graphs and show that every optimal (conditional) matching preclusion set is trivial.

Strong matching preclusion under the conditional fault model

Strong matching preclusion that additionally permits more destructive vertex faults in a graph [J.-H. Park, I. Ihm, Strong matching preclusion, Theoretical Computer Science 412 (2011) 6409–6419] is an extended form of the original matching preclusion that assumes only edge faults [R.C. Brigham, F. Harary, E.C. Violin, J. Yellen, Perfect-matching preclusion, Congressus Numerantium 174 (2005) 185–192]. In this paper, we study the problem of strong matching preclusion under the condition that no isolated vertex is created as a result of faults. After briefly discussing some fundamental classes of graphs in the point of the conditional matching preclusion, we establish the conditional strong matching preclusion number for the class of restricted hypercube-like graphs, which include most nonbipartite hypercube-like networks found in the literature.