TY - JOUR

T1 - Structural properties of resonance graphs of plane elementary bipartite graphs

AU - Che, Zhongyuan

N1 - Funding Information:
This paper is supported by the Penn State Research Development Grant (RDG). The author would like to thank the referees for their helpful comments.
Funding Information:
This paper is supported by the Penn State Research Development Grant (RDG) . The author would like to thank the referees for their helpful comments.
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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.

AB - 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.

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U2 - 10.1016/j.dam.2018.03.065

DO - 10.1016/j.dam.2018.03.065

M3 - Article

AN - SCOPUS:85045095212

VL - 247

SP - 102

EP - 110

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -