Matching preclusion and conditional matching preclusion for regular 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. For many interconnection networks, the optimal sets are precisely those induced by a single vertex. Recently, the conditional matching preclusion number of a 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 neither perfect matchings nor almost-perfect matchings. In this paper, we prove general results regarding the matching preclusion number and the conditional matching preclusion number as well as the classification of their respective optimal sets for regular graphs. We then use these general results to study the problems for Cayley graphs generated by 2-trees and the hyper Petersen networks.     

Anti-forcing spectrum of any cata-condensed hexagonal system is continuous

The anti-forcing number of a perfect matching M of a graph G is the minimal number of edges not in M whose removal makes M a unique perfect matching of the resulting graph. The anti-forcing spectrum of G is the set of anti-forcing numbers over all perfect matchings of G: In this paper, we prove that the anti-forcing spectrum of any cata-condensed hexagonal system is continuous, that is, it is a finite set of consecutive integers.     

Total matchings and total coverings of graphs

In graph theory, the related problems of deciding when a set of vertices or a set of edges constitutes a maximum matching or a minimum covering have been extensively studied. In this paper we generalize these ideas by defining total matchings and total coverings, and show that these sets, whose elements in general consist of both vertices and edges, provide a way to unify these concepts. Parameters denoting the maximum and the minimum cardinality of these sets are introduced and upper and lower bounds depending only on the order of the graph are obtained for the number of elements in arbitrary total matchings and total coverings. Precise values of all the parameters are found for several general classes of graphs, and these are used to establish the sharpness of most of the bounds. In addition, variations of some well known equalities due to Gallai relating covering and matching numbers are obtained.     

On forcing matching number of boron-nitrogen fullerene graphs

Let G be a graph that admits a perfect matching M. A forcing set S for a perfect matching M is a subset of M such that it is contained in no other perfect matchings of G. The smallest cardinality of forcing sets of M is called the forcing number of M. Computing the minimum forcing number of perfect matchings of a graph is an NP-complete problem. In this paper, we consider boron-nitrogen (BN) fullerene graphs, cubic 3-connected plane bipartite graphs with exactly six square faces and other hexagonal faces. We obtain the forcing spectrum of tubular BN-fullerene graphs with cyclic edge-connectivity 3. Then we show that all perfect matchings of any BN-fullerene graphs have the forcing number at least two. Furthermore, we mainly construct all seven BN-fullerene graphs with the minimum forcing number two.     

Perfect matchings in regular bipartite graphs

P. Hall, [2], gave necessary and sufficient conditions for a bipartite graph to have a perfect matching. Koning, [3], proved that such a graph can be decomposed intok edge-disjoint perfect matchings if and only if it isk-regular. It immediately follows that in ak-regular bipartite graphG, the deletion of any setS of at mostk – 1 edges leaves intact one of those perfect matchings. However, it is not known what happens if we delete more thank – 1 edges. In this paper we give sufficient conditions so that by deleting a setS ofk + r edgesr 0, stillG – S has a perfect matching. Furthermore we prove that our result, in some sense, is best possible.     

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.     

Perfect Matchings of Regular Bipartite Graphs

Let G be a regular bipartite graph and . We show that there exist perfect matchings of G containing both, an odd and an even number of edges from X if and only if the signed graph , that is a graph G with exactly the edges from X being negative, is not equivalent to . In fact, we prove that for a given signed regular bipartite graph with minimum signature, it is possible to find perfect matchings that contain exactly no negative edges or an arbitrary one preselected negative edge. Moreover, if the underlying graph is cubic, there exists a perfect matching with exactly two preselected negative edges. As an application of our results we show that each signed regular bipartite graph that contains an unbalanced circuit has a 2‐cycle‐cover such that each cycle contains an odd number of negative edges.     

Graphs with independent perfect matchings

A graph with at least two vertices is matching covered if it is connected and each edge lies in some perfect matching. A matching covered graph G is extremal if the number of perfect matchings of G is equal to the dimension of the lattice spanned by the set of incidence vectors of perfect matchings of G. We first establish several basic properties of extremal matching covered graphs. In particular, we show that every extremal brick may be obtained by splicing graphs whose underlying simple graphs are odd wheels. Then, using the main theorem proved in 2 and 3 , we find all the extremal cubic matching covered graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 19–50, 2005     

On some structural properties of generalized fullerene graphs with 13 pentagonal faces

It is shown that every generalized fullerene graph G with 13 pentagons is 2-extendable, a brick, and cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. New lower bound on the number of perfect matchings in such graphs are also established.     

Matchings in hypergraphs of large minimum degree

It is well known that every bipartite graph with vertex classes of size n whose minimum degree is at least n/2 contains a perfect matching. We prove an analog of this result for hypergraphs. We also prove several related results that guarantee the existence of almost perfect matchings in r‐uniform hypergraphs of large minimum degree. Our bounds on the minimum degree are essentially best possible. © 2005 Wiley Periodicals, Inc. J Graph Theory 51: 269–280, 2006     

Perfect Matchings of Polyomino Graphs

This paper gives necessary and sufficient conditions for a polyomino graph to have a perfect matching and to be elementary, respectively. As an application, we can decompose a non-elementary polyomino with perfect matchings into a number of elementary subpolyominoes so that the number of perfect matchings of the original non-elementary polyomino is equal to the product of those of the elementary subpolyominoes.     

Optimum matching forests II: General weights

Matching forests generalize branchings in a directed graph and matchings in an undirected graph. We present an efficient algorithm, the PMF Algorithm, for the problem: given a mixed graphG and a real weight on each of its edges, find a perfect matching forest of maximum weight-sum. The PMF Algorithm proves the sufficiency of a linear system which definesP ⁼ (G) andP(G), the convex hull of incidence vectors of perfect matching forests and matching forests respectively ofG. The algorithm also provides a generalization of Tutte's theorem on the existence of perfect matchings in an undirected graph.Research partially supported by a N.R.C. of Canada Postdoctorate Fellowship.     

Forcing matching numbers of fullerene graphs

The forcing number or the degree of freedom of a perfect matching M of a graph G is the cardinality of the smallest subset of M that is contained in no other perfect matchings of G. In this paper we show that the forcing numbers of perfect matchings in a fullerene graph are not less than 3 by applying the 2-extendability and cyclic edge-connectivity 5 of fullerene graphs obtained recently, and Kotzig’s classical result about unique perfect matching as well. This lower bound can be achieved by infinitely many fullerene graphs.     

On bipartite matchings of minimum density

We study the complexity of bipartite matchings between points on two horizontal lines when the density (i.e., the number of edges of the matching that cross any vertical cut) is to be minimized. We show that finding density-minimizing matchings is considerably harder than finding weight-minimizing matchings by proving that the problem is NP-hard for general bipartite graphs. For a number of variants we present a new combinatorial characterization of the optimal density that leads to efficient and elegant algorithms. The problem of finding a matching of minimum density has applications in the area of circuit layout.     

Matching structure and the matching lattice

The matching polyhedron, i.e., the convex hull of (incidence vectors of) perfect matchings of a graph was characterized by Edmonds; this result is the key to a large part of polyhedral combinatorics and is used in many combinatorial algorithms. The linear hull of perfect matchings was characterized by Naddef, and by Edmonds, Lovász, and Pulleyblank. In this paper we describe the lattice generated by these vectors, i.e., the set of all integer linear combinations of perfect matchings. It turns out that the Petersen graph is, in a sense, the only difficult example. Our results also imply a characterization of the linear hull of perfect matchings over fields of characteristic different from 0. The main method is a decomposition theory developed by Kotzig, Lovász, and Plummer, which breaks down every graph into a number of graphs called bricks with very good matching properties. The number of Petersen graphs among these bricks will turn out to be an essential parameter of the matching lattice. Some refinements of the decomposition theory are also given. Among others, we show that the list of bricks obtained during the decomposition procedure is independent of the special choices made during the procedure.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

一类变换图的连通性

本文定义了一类由给定的一个３－正则平面偶图的全体完美匹配所构成的变换图，并证明了该变换图是连通的．由此可得出结论：从任一给定的３－正则平面偶图的完美匹配出发，通过一种所谓的旋转运算，就可以生成全部其它的完美匹配．     

Unions of perfect matchings in cubic graphs

We show that any cubic bridgeless graph with m edges contains two perfect matchings that cover at least 3m/5 and three perfect matchings that cover at least 27m/35 of its edges.