Plane Elementary Bipartite Graph Research Articles

Resonance Graphs on Perfect Matchings of Graphs on Surfaces

Let G be a graph embedded in a surface and let be the set of all faces of G. For a given subset $$\mathcal F$$ of even faces (faces bounded by an even cycle), the resonance graph of G with respect to $${\mathcal {F}}$$ , denoted by $$R(G;\mathcal F)$$ , is a graph such that its vertex set is the set of all perfect matchings of G and two vertices $$M_1$$ and $$M_2$$ are adjacent if and only if the symmetric difference $$M_1\oplus M_2$$ is a cycle bounding some face in $${\mathcal {F}}$$ . It has been shown that if G is a plane elementary bipartite graph, the resonance graph of G with respect to the set of all inner faces is isomorphic to the covering graph of a distributive lattice. However, this result does not hold in general for plane graphs G. The structure properties of resonance graphs in general remain unknown. In this paper, we show that every connected component of the resonance graph of a graph G on a surface with respect to a given even-face set can always be embedded into a hypercube as an induced subgraph. Further, we show that the Clar covering polynomial of G with respect to $${\mathcal {F}}$$ is equal to the cube polynomial of the resonance graph $$R(G;{\mathcal {F}})$$ .

Peripheral Convex Expansions of Resonance Graphs

A perfect matching of a graph is a set of independent edges that covers all vertices of the graph. A bipartite graph is elementary if and only if it is connected and each edge is contained in a perfect matching. The resonance graph of a plane bipartite graph G is a graph whose vertices are perfect matchings of G, and two perfect matchings are adjacent if edges contained in their union but not intersection form a cycle surrounding a finite face of G. It is well known that the resonance graph of a plane elementary bipartite graph is a median graph. Median graphs form an important subclass of partial cubes and have a wide range of applications. The most important structural characterization of a median graph is the Mulder’s convex expansion theorem. Fibonacci cubes used in network designs are median graphs and can be obtained from an edge by a sequence of peripheral convex expansions. Plane bipartite graphs whose resonance graphs are Fibonacci cubes were studied by Klavžar et al. first for their applications in chemistry and completely characterized by Zhang et al. later. Motivated from their work, we characterize all plane bipartite graphs whose resonance graphs can be constructed from an edge by a sequence of peripheral convex expansions.

Cube-free resonance graphs

Let G be a plane elementary bipartite graph with more than two vertices. Then its resonance graph Z(G) is a median graph and the set M(G) of all perfect matchings of G with a specific partial order is a finite distributive lattice. In this paper, we prove that Z(G) is cube-free if and only if it can be obtained from an edge by a sequence of convex path expansions with respect to a reducible face decomposition of G. As a corollary, a structure characterization is provided for G whose Z(G) is cube-free. Furthermore, Z(G) is cube-free if and only if the Clar number of G is at most two, and sharp lower bounds on the number of perfect matchings of G can be expressed by the number of finite faces of G and the number of Clar formulas of G. It is known that a cube-free median graph is not necessarily planar. Using the lattice structure on M(G), we show that Z(G) is cube-free if and only if Z(G) is planar if and only if M(G) is an irreducible sublattice of m×n. We raise a question on how to characterize irreducible sublattices of m×n that are M(G).

Plane Elementary Bipartite Graphs with Forcing or Anti-forcing Edges

Let G be a plane elementary bipartite graph with more than one finite face. We first characterize forcing edges and anti-forcing edges of G in terms of handles and forcing faces. We then introduce the concept of a nice pair of faces of G, which allows us to provide efficient algorithms to construct plane minimal elementary bipartite graphs from G. We show that if $$G'$$ is a minimal elementary spanning subgraph of G, then all vertices of a forcing face of G are contained in a forcing face of $$G'$$ , and any forcing face of G is a forcing face of $$G'$$ if it is still a face of $$G'$$ . Furthermore, any forcing edge (resp., anti-forcing edge) of G is still a forcing edge (resp., an anti-forcing edge) of $$G'$$ if it is still an edge of $$G'$$ . We conclude the paper with the co-existence property of forcing edges and anti-forcing edges for those plane minimal elementary bipartite graphs that remain 2-connected when any handle of length one (if exists) is deleted.

Structural properties of resonance graphs of plane elementary bipartite graphs

In this paper, we investigate some structural properties of resonance graphs of plane elementary bipartite graphs using Djoković – Winkler relation Θ and structural characterizations of a median graph. Let G be a plane elementary bipartite graph. It is known that its resonance graph Z(G) is a median graph. We first provide properties for Θ-classes of the edge set of Z(G). As a corollary, Z(G) cannot be a nontrivial Cartesian product of median graphs, which is equivalent to a result given by Zhang et al. that the distributive lattice on the set of perfect matchings of G is irreducible. We then present a decomposition structure on Z(G) with respect to a reducible face s of G. As an application, we give a necessary and sufficient condition on when Z(G) can be obtained from Z(H) by a peripheral convex expansion with respect to a reducible face s of G, where H is the subgraph of G obtained by removing all internal vertices (if exist) and edges on the common periphery of s and G. Furthermore, we show that Z(G) can be obtained from Z(H) by adding one pendent edge with the face-label s if and only if s is a forcing face of G such that both s and the infinite face of G are M-resonant for a degree-1 vertex M of Z(G). Our results generalize the peripheral convex expansion structure on Z(G) given by Klavžar et al. for the case when G is a catacondensed even ring system.

Extremal anti-forcing numbers of perfect matchings of graphs

The anti-forcing number of a perfect matching M of a graph G is the minimal number of edges not in M whose removal to make M as a unique perfect matching of the resulting graph. The set of anti-forcing numbers of all perfect matchings of G is the anti-forcing spectrum of G. In this paper, we characterize the plane elementary bipartite graph whose minimum anti-forcing number is one. We show that the maximum anti-forcing number of a graph is at most its cyclomatic number. In particular, we characterize the graphs with the maximum anti-forcing number achieving the upper bound, such extremal graphs are a class of plane bipartite graphs. Finally, we determine the anti-forcing spectrum of an even polygonal chain in linear time.

Non-matchable distributive lattices

Based on an acyclic orientation of the Z-transformation graph, a finite distributive lattice (FDL for short) M(G) is established on the set of all 1-factors of a plane (weakly) elementary bipartite graph G. For an FDL L, if there exists a plane bipartite graph G such that L is isomorphic to M(G), then L is called a matchable FDL. A natural question arises: Is every FDL a matchable FDL? In this paper we give a negative answer to this question. Further, we obtain a series of non-matchable FDLs by characterizing sub-structures of matchable FDLs with cut-elements.

Forcing faces in plane bipartite graphs (II)

The concept of forcing faces of a plane bipartite graph was first introduced in Che and Chen (2008) [3] [Z. Che, Z. Chen, Forcing faces in plane bipartite graphs, Discrete Mathematics 308 (2008) 2427–2439], which is a natural generalization of the concept of forcing hexagons of a hexagonal system introduced in Che and Chen (2006) [2] [Z. Che and Z. Chen, Forcing hexagons in hexagonal systems, MATCH Commun. Math. Comput. Chem. 56 (2006) 649–668]. In this paper, we further extend this concept from finite faces to all faces (including the infinite face) as follows: A face s (finite or infinite) of a 2-connected plane bipartite graph G is called a forcing face if the subgraph G−V(s) obtained by removing all vertices of s together with their incident edges has exactly one perfect matching.For a plane elementary bipartite graph G with more than two vertices, we give three necessary and sufficient conditions for G to have all faces forcing. We also give a new necessary and sufficient condition for a finite face of G to be forcing in terms of bridges in the Z-transformation graph Z(G) of G. Moreover, for the graphs G whose faces are all forcing, we obtain a characterization of forcing edges in G by using the notion of handle, from which a simple counting formula for the number of forcing edges follows.

1-factors and characterization of reducible faces of plane elementary bipartite graphs

As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekule structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the re- moval of the internal vertices and edges of the path that is the intersection of f and the outer cycle of G results in an elementary graph. We characterize the reducible faces of a plane elementary bipartite graph. This result gen- eralizes the characterization of reducible faces of an elementary benzenoid graph.

Resonance Graphs and a Binary Coding for the 1-Factors of Benzenoid Systems

Applying the recently obtained distributive lattice structure on the set of 1-factors, we show that the resonance graphs of any benzenoid systems G, as well as of general plane (weakly) elementary bipartite graphs, are median graphs and thus extend greatly Klavžar et al.'s result. The n-dimensional vectors of nonnegative integers as a labelling for the 1-factors of G with n inner faces are described. The labelling preserves the partial ordering of the above-mentioned lattice and can be transformed into a binary coding for the 1-factors. A simple criterion for such a labelling being binary is given. In particular, Klavžar et al.'s algorithm is modified to generate this binary coding for the 1-factors of a cata-condensed benzenoid system.

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.

Z-transformation graphs of maximum matchings of plane bipartite graphs

Let G be a plane bipartite graph. The Z-transformation graph Z(G) and its orientation Z→(G) on the maximum matchings of G are defined. If G has a perfect matching, Z(G) and Z→(G) are the usual Z-transformation graph and digraph. If G has neither isolated vertices nor perfect matching, then Z(G) is not connected. This paper mainly shows that some basic results for Z-transformation graph (digraph) of a plane elementary bipartite graph still hold for every nontrivial component of Z(G)(Z→(G)). In particular, by obtaining a result that every shortest path of Z(G) from a source of Z→(G) corresponds to a directed path of Z→(G), we show that every nontrivial component of Z→(G) has exactly one source and one sink. Accordingly, it follows that the block graph of every nontrivial component of Z(G) is a path. In addition, we give a simple characterization for a maximum matching of G being a cut-vertex of Z(G).

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D( G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D( G) is a directed forest and every component is an in-tree with one root; if T is a component of D( G), the reversions of all orientations in T induce a component of D( G), denoted by T −, thus ( T, T −) is called a pair of in-trees of D( G); G is Eulerian if and only if D( G) has an odd number of components (all Eulerian orientations of G induce the same component of D( G)); the width and height of T are equal to that of T −, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D( G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.

Plane elementary bipartite graphs

A connected graph is called elementary if the union of all perfect matchings forms a connected subgraph. In this paper we mainly study various properties of plane elementary bipartite graphs so that many important results previously obtained for hexagonal systems are treated in a unified way. Firstly, we show that a plane bipartite graph G is elementary if and only if the boundary of each face (including the infinite face) is an alternating cycle with respect to some perfect matching of G. For a plane bipartite graph G all interior vertices of which are of the same degree, a stronger result is obtained; namely, G is elementary if and only if the boundary of the infinite face of G is an alternating cycle with respect to some perfect matching of G. Second, the concept of the Z-transformation graph Z( G) of a hexagonal system G (whose vertices represent the perfect matchings of G) is extended to a plane bipartite graph G and some results analogous to those for hexagonal systems are obtained. A peripheral face f of G is called reducible if the removal of the internal vertices and edges of the path that is the intersection of f and the exterior face of G results in a plane elementary bipartite graph. Thirdly, we obtain the reducible face decomposition for plane elementary bipartite graphs. Furthermore, sharp upper and lower bounds for the number of reducible faces are derived. Conversely, we can construct any plane elementary bipartite graphs by adding new peripheral faces one by one. As applications of this approach, we give simple construction methods for several types of plane elementary bipartite graphs G that contain a forcing edge (which belongs to exactly one perfect matching of G) and whose Z-transformation graphs Z( G) contain vertices of degree one.

Total Z-transformation graphs of perfect matching of plane bipartite graphs

Abstract The total Z-transformation graph and digraph of perfect matchings of a plane bipartite graph are defined. For a plane elementary bipartite graph it is shown that its total Z-transformation graph and digraph are 2-connected and strongly connected respectively. As an immediate consequence, we have that a plane bipartite graph is elementary if and only if its total Z-transformation digraph is strongly connected.