Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8941806 | Discrete Applied Mathematics | 2018 | 9 Pages |
Abstract
In this paper, we investigate some structural properties of resonance graphs of plane elementary bipartite graphs using DjokoviÄ - Winkler relation Î 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 Î-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.
Related Topics
Physical Sciences and Engineering
Computer Science
Computational Theory and Mathematics
Authors
Zhongyuan Che,