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.     

Unicycle graphs and uniquely restricted maximum matchings

A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. G is a unicycle graph if it owns only one cycle. Golumbic, Hirst and Lewenstein observed that for a tree or a graph with only odd cycles the size of a maximum uniquely restricted matching is equal to the matching number of the graph. In this paper we characterize unicycle graphs enjoying this equality. Moreover, we describe unicycle graphs with only uniquely restricted maximum matchings. Using these findings, we show that unicycle graphs having only uniquely restricted maximum matchings can be recognized in polynomial time.     

Forcing faces in plane bipartite graphs

Let Ω denote the class of connected plane bipartite graphs with no pendant edges. A finite face s of a graph G∈Ω is said to be a forcing face of G if the subgraph of G obtained by deleting all vertices of s together with their incident edges has exactly one perfect matching. This is a natural generalization of the concept of forcing hexagons in a hexagonal system introduced in Che and Chen [Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (3) (2006) 649-668]. We prove that any connected plane bipartite graph with a forcing face is elementary. We also show that for any integers n and k with n?4 and n?k?0, there exists a plane elementary bipartite graph such that exactly k of the n finite faces of G are forcing. We then give a shorter proof for a recent result that a connected cubic plane bipartite graph G has at least two disjoint M-resonant faces for any perfect matching M of G, which is a main theorem in the paper [S. Bau, M.A. Henning, Matching transformation graphs of cubic bipartite plane graphs, Discrete Math. 262 (2003) 27-36]. As a corollary, any connected cubic plane bipartite graph has no forcing faces. Using the tool of Z-transformation graphs developed by Zhang et al. [Z-transformation graphs of perfect matchings of hexagonal systems, Discrete Math. 72 (1988) 405-415; Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291-311], we characterize the plane elementary bipartite graphs whose finite faces are all forcing. We also obtain a necessary and sufficient condition for a finite face in a plane elementary bipartite graph to be forcing, which enables us to investigate the relationship between the existence of a forcing edge and the existence of a forcing face in a plane elementary bipartite graph, and find out that the former implies the latter but not vice versa. Moreover, we characterize the plane bipartite graphs that can be turned to have all finite faces forcing by subdivisions.     

On Unicyclic Graphs with Uniquely Restricted Maximum Matchings

A graph is called unicyclic if it owns only one cycle. A matching M is called uniquely restricted in a graph G if it is the unique perfect matching of the subgraph induced by the vertices that M saturates. Clearly, μ _r (G) ≤ μ(G), where μ _r (G) denotes the size of a maximum uniquely restricted matching, while μ(G) equals the matching number of G. In this paper we study unicyclic bipartite graphs enjoying μ _r (G) = μ(G). In particular, we characterize unicyclic bipartite graphs having only uniquely restricted maximum matchings. Finally, we present some polynomial time algorithms recognizing unicyclic bipartite graphs with (only) uniquely restricted maximum matchings.     

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p≥3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.     

Bounds on the forcing numbers of bipartite graphs

The forcing number 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 matching of G. In this paper, we demonstrate several techniques to produce upper bounds on the forcing number of bipartite graphs. We present a simple method of showing that the maximum forcing number on the 2m×2n rectangle is mn, and show that the maximum forcing number on the 2m×2n torus is also mn. Further, we investigate the lower bounds on the forcing number and determine the conditions under which a previously formulated lower bound is sharp; we provide an example of a family of graphs for which it is arbitrarily weak.     

Matching graphs

The matching graph M(G) of a graph G is that graph whose vertices are the maximum matchings in G and where two vertices M₁ and M₂ of M(G) are adjacent if and only if |M₁ − M₂| = 1. When M(G) is connected, this graph models a metric space whose metric is defined on the set of maximum matchings in G. Which graphs are matching graphs of some graph is not known in general. We determine several forbidden induced subgraphs of matching graphs and add even cycles to the list of known matching graphs. In another direction, we study the behavior of sequences of iterated matching graphs. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 73–86, 1998     

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     

The graph of perfect matching polytope and an extreme problem

For graph G, its perfect matching polytope Poly(G) is the convex hull of incidence vectors of perfect matchings of G. The graph corresponding to the skeleton of Poly(G) is called the perfect matching graph of G, and denoted by PM(G). It is known that PM(G) is either a hypercube or hamilton connected [D.J. Naddef, W.R. Pulleyblank, Hamiltonicity and combinatorial polyhedra, J. Combin. Theory Ser. B 31 (1981) 297-312; D.J. Naddef, W.R. Pulleyblank, Hamiltonicity in (0-1)-polytope, J. Combin. Theory Ser. B 37 (1984) 41-52]. In this paper, we give a sharp upper bound of the number of lines for the graphs G whose PM(G) is bipartite in terms of sizes of elementary components of G and the order of G, respectively. Moreover, the corresponding extremal graphs are constructed.     

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.     

2-resonance of plane bipartite graphs and its applications to boron-nitrogen fullerenes

A set H of disjoint faces of a plane bipartite graph G is a resonant pattern if G has a perfect matching M such that the boundary of each face in H is an M-alternating cycle. An elementary result was obtained [Discrete Appl. Math. 105 (2000) 291-311]: a plane bipartite graph is 1-extendable if and only if every face forms a resonant pattern. In this paper we show that for a 2-extendable plane bipartite graph, any pair of disjoint faces form a resonant pattern, and the converse does not necessarily hold. As an application, we show that all boron-nitrogen (B-N) fullerene graphs are 2-resonant, and construct all the 3-resonant B-N fullerene graphs, which are all k-resonant for any positive integer k. Here a B-N fullerene graph is a plane cubic graph with only square and hexagonal faces, and a B-N fullerene graph is k-resonant if any disjoint faces form a resonant pattern. Finally, the cell polynomials of 3-resonant B-N fullerene graphs are computed.     

Construction for bicritical graphs and k-extendable bipartite graphs

A graph G is said to be bicritical if G-u-v has a perfect matching for every choice of a pair of points u and v. Bicritical graphs play a central role in decomposition theory of elementary graphs with respect to perfect matchings. As Plummer pointed out many times, the structure of bicritical graphs is far from completely understood. This paper presents a concise structure characterization on bicritical graphs in terms of factor-critical graphs and transversals of hypergraphs. A connected graph G with at least 2k+2 points is said to be k-extendable if it contains a matching of k lines and every such matching is contained in a perfect matching. A structure characterization for k-extendable bipartite graphs is given in a recursive way. Furthermore, this paper presents an O(mn) algorithm for determining the extendability of a bipartite graph G, the maximum integer k such that G is k-extendable, where n is the number of points and m is the number of lines in G.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

Distance-restricted matching extension in planar triangulations

A graph G is said to have property E(m,n) if it contains a perfect matching and for every pair of disjoint matchings M and N in G with |M|=m and |N|=n, there is a perfect matching F in G such that M⊆F and N∩F=0?. In a previous paper (Aldred and Plummer 2001) [2], an investigation of the property E(m,n) was begun for graphs embedded in the plane. In particular, although no planar graph is E(3,0), it was proved there that if the distance among the three edges is at least two, then they can always be extended to a perfect matching. In the present paper, we extend these results by considering the properties E(m,n) for planar triangulations when more general distance restrictions are imposed on the edges to be included and avoided in the extension.     

Perfect matchings after vertex deletions

This paper considers some classes of graphs which are easily seen to have many perfect matchings. Such graphs can be considered robust with respect to the property of having a perfect matching if under vertex deletions (with some mild restrictions), the resulting subgraph continues to have a perfect matching. It is clear that you can destroy the property of having a perfect matching by deleting an odd number of vertices, by upsetting a bipartition or by deleting enough vertices to create an odd component. One class of graphs we consider is the m×m lattice graph (or grid graph) for m even. Matchings in such grid graphs correspond to coverings of an m×m checkerboard by dominoes. If in addition to the easy conditions above, we require that the deleted vertices be apart, the resulting graph has a perfect matching. The second class of graphs we consider is a k-fold product graph consisting of k copies of a given graph G with the ith copy joined to the i+1st copy by a perfect matching joining copies of the same vertex. We show that, apart from some easy restrictions, we can delete any vertices from the kth copy of G and find a perfect matching in the product graph with k suitably large.     

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.     

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.