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A characterization of the resonance graph of an outerplane bipartite graph
Discrete Applied Mathematics xxx (xxxx) xxxx
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A characterization of the resonance graph of an outerplane bipartite graph Zhongyuan Che Department of Mathematics, Penn State University, Beaver Campus, Monaca, PA 15061, USA
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Article history: Received 6 June 2018 Received in revised form 24 October 2018 Accepted 26 November 2018 Available online xxxx Keywords: Djoković–Winkler relation Θ Median graph Outerplane bipartite graph Peripheral convex expansion Reducible face Resonance graph Z -transformation graph
a b s t r a c t Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. It is known that R(G) is a median graph. Assume that s is a reducible face of G and H is the subgraph of G obtained by removing all internal vertices (if exist) and edges on the common periphery of s and G. We show that R(G) can be obtained from R(H) by a peripheral convex expansion. As an application, we prove that Θ (R(G)) is a tree and isomorphic to the inner dual of G, where Θ (R(G)) is the induced graph on the Djoković–Winkler relation Θ -classes of R(G). © 2018 Elsevier B.V. All rights reserved.
1. Introduction Let G be a plane bipartite graph with a perfect matching. The resonance graph of G, denoted by R(G), is the graph whose vertices are the perfect matchings of G, and two vertices M1 and M2 of R(G) are adjacent if and only if their symmetric difference is the periphery of a finite face s of G, and we say that the edge M1 M2 has the face-label s. The resonance graph of G is also called the Z -transformation graph of G. See Fig. 1 for the resonance graph R(G) of a 2-connected outerplane bipartite graph G. A peripheral face s of a plane elementary bipartite graph G is called reducible if the subgraph H of G obtained by removing all internal vertices (if exist) and edges on the common periphery of s and G is elementary. We gave a necessary and sufficient condition on when R(G) can be obtained from R(H) by a peripheral convex expansion with respect to a reducible face s of G in [1]. A peripheral expansion structure for the resonance graph of a catacondensed benzenoid graph (also called catacondensed hexagonal graph) was given in [3], and a peripheral convex expansion structure for the resonance graph of a catacondensed even ring system was given in [4]. A characterization of the resonance graph of a catacondensed hexagonal graph was presented in [6] in terms of the induced graph on the Djoković–Winkler relation Θ -classes of its resonance graph. In this paper, let G be a 2-connected outerplane bipartite graph. We first show that if s is a reducible face of G, then s has common edge(s) with exactly one finite face of G. By the decomposition structure of R(G) from [1], it follows that R(G) can be obtained from R(H) by a peripheral convex expansion. We conclude the paper with a characterization of R(G) using the tool of Djoković–Winkler relation Θ : The induced graph Θ (R(G)) on the Θ -classes of R(G) is a tree and isomorphic to the inner dual of G. This generalizes the corresponding results given in [3,4] and [6]. E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2018.11.032 0166-218X/© 2018 Elsevier B.V. All rights reserved.
Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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Z. Che / Discrete Applied Mathematics xxx (xxxx) xxxx
Fig. 1. A 2-connected outerplane bipartite graph G with a perfect matching M ′ . R(G) is the resonance graph of G and Θ (R(G)) is the induced graph on
Θ -classes of R(G).
2. Preliminaries A perfect matching (or, 1-factor) of a graph is a set of pairwise disjoint edges of the graph that covers all its vertices. Let M be a perfect matching of a graph G. An M-alternating cycle (resp., M-alternating path) is a cycle (resp., path) of G whose edges are alternatively in and off M. A plane graph is a graph in the plane where any two edges are either disjoint or meet only at a common end vertex. Each interior region of a plane graph G is called a finite face (or, an inner face) of G, and the exterior region of G is called the infinite face (or, exterior face) of G. Two faces (one can be the infinite face) of a plane graph G are adjacent if their peripheries have common edges. A finite face s of G is called a peripheral face if it is adjacent to the infinite face of G. The inner dual of a plane graph G is a plane graph whose vertex set is in 1–1 correspondence with the set of finite faces of G, and two vertices are adjacent if the corresponding finite faces of G are adjacent. If a plane graph G is 2-connected, then the periphery of any face of G is a cycle. The periphery of a finite face s of G is denoted by ∂ s. The periphery of the infinite face of G is denoted by ∂ G, which is referred as the periphery (or, boundary) of G. A face (including the infinite face) of G is called M-resonant if its periphery is an M-alternating cycle for a perfect matching M of G. A vertex of a plane graph is called an exterior vertex if it is located on the periphery of the graph, and an interior vertex otherwise. An even ring system is a 2-connected plane bipartite graph whose interior vertices are degree-3 vertices and exterior vertices are degree-2 or degree-3 vertices. An outerplane graph is a plane graph whose vertices are all exterior vertices. A catacondensed even ring system is an outerplane graph that is also an even ring system. A catacondensed benzenoid graph is a catacondensed even ring system whose finite faces are hexagons. A bipartite graph G is elementary if and only if it is connected and each edge is contained in some perfect matching of G [5]. We assume that all vertices of a bipartite graph are colored white and black such that adjacent vertices have different colors. A bipartite graph G is elementary if and only if it has a bipartite ear decomposition G = e + P1 + P2 + · · · + Pn starting from an edge G0 = e, and Gi is obtained from Gi−1 = e + P1 +· · ·+ Pi−1 by adding the ith ear Pi of odd length such that Pi joins two vertices in different colors of Gi−1 and Pi has no internal vertices in common with the vertices of Gi−1 [5]. A bipartite ear decomposition of a plane elementary bipartite graph G is called a reducible face decomposition (briefly, RFD) [8] if G1 is a boundary of a finite face s1 of G, and the ith ear Pi lies in the exterior of Gi−1 such that Pi and a part of the periphery of Gi−1 surround a finite face si of G for all 2 ≤ i ≤ n. For any RFD(G1 , G2 , . . . , Gn (= G)) of G associated with the face sequence si (1 ≤ i ≤ n), each si is a reducible face of Gi for 2 ≤ i ≤ n. Note that any 2-connected outerplane bipartite graph is elementary and so has a reducible face decomposition. For example, the 2-connected outerplane bipartite graph G in Fig. 1 has a RFD(G0 , G1 , . . . , G6 (= G)) associated with the face sequence si (0 ≤ i ≤ 6). The interval between two vertices u and v of a graph G is the set of all vertices on shortest paths between u and v in G, and denoted by IG (u, v ). A connected graph G is called a median graph if IG (u, v ) ∩ IG (u, w ) ∩ IG (v, w ) is a unique vertex for every triple of its vertices u, v, w . The Djoković–Winkler relation Θ plays an important role on the structural characterization of median graphs. Two edges uv and xy of a graph G are said to be in relation Θ , denoted by uv Θ xy, if dG (u, x) + dG (v, y) ̸ = dG (u, y) + dG (v, x). If G is a median graph, then the relation Θ defines an equivalence relation on the edge set of G [2]. Let G be a plane elementary bipartite graph and R(G) be its resonance graph. Then R(G) is a median graph [7]. Let xy be an edge of R(G) and Fxy = {e ∈ E(R(G)) | eΘ xy} be the set of all edges in relation Θ with xy in R(G). By Proposition 3.2 in [1], all edges in Fxy have the same face-label. On the other hand, two edges with the same face-label can be in different Θ -classes of R(G). The induced graph Θ (R(G)) on the Θ -classes of R(G) is a graph whose vertex set is the set of Θ -classes, and two vertices E and F of Θ (R(G)) are adjacent if R(G) has two incident edges e ∈ E and f ∈ F such that e and f are not contained in a common 4-cycle of R(G). See Fig. 1. Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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Let M be a perfect matching of a graph G. A path P of G is weakly M-augmenting [1] if it satisfies one of the following conditions: (i) P has length 1 and the single edge of P is not contained in M, (ii) P is an M-alternating path such that its two end edges are not contained in M. A quick remark is that a weakly M-augmenting path is different from an M-augmenting path defined in [5], where the path has length > 1 and its two end vertices are not covered by a non-perfect matching M. (There is a typo in [1] for the remark where ‘‘non-’’ was missed.) A path in a graph is called a handle if both its two end vertices have degree > 2 but all its internal vertices (if any) are degree-2 vertices of the graph. Let s be a reducible face of a plane elementary bipartite graph G. Then the common periphery of s and G is an odd length handle P [1]. By Proposition 4.1 in [1], the set of all perfect matchings M(G) of G has a partition M(G) = M(G; P − ) ∪ M(G; P + ), where M(G; P − ) is the set of perfect matchings M of G such that P is weakly M-augmenting, M(G; P + ) is the set of perfect matchings M of G such that P is not weakly M-augmenting. Let G be a graph with the vertex set V (G). An induced subgraph of G generated by a subset W ⊆ V (G) is denoted by ⟨W ⟩. An induced subgraph ⟨W ⟩ of G is called a convex subgraph if all shortest paths between u and v are contained in ⟨W ⟩ for any u, v ∈ W . Let T be a convex subgraph of G such that V (T ) ⊂ V (G). The peripheral convex expansion of G with respect to T , denoted by pce(G, T ), is the graph obtained from G by the following procedure [2]. 1. Replace each vertex v of T by an edge v1 v2 . 2. Insert edges between v1 and the neighbors of v in V (G) \ V (T ). 3. Insert the edges u1 v1 and u2 v2 whenever u, v of T are adjacent in G. A periphery expansion pe(G; T ) where T is just a subgraph of G can be obtained as the same procedure for pce(G; T ) by dropping the convex condition on T . A periphery expansion structure on R(G) was given in [3] when G is a catacondensed benzenoid graph, and a periphery convex expansion structure on R(G) was given in [4] when G is a catacondensed even ring system. We provided a necessary and sufficient condition on when R(G) can be obtained by a peripheral convex expansion with respect to a reducible face s of G in [1]. Theorem 2.1 ([1]). Let G be a plane elementary bipartite graph and R(G) be its resonance graph. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then R(G) can be obtained from R(H) by a peripheral convex expansion if and only if s is M-resonant for any perfect matching M in M(G; P + ). Moreover, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class of R(G) with the face-label s. 3. Main results Proposition 3.1. Let G be a 2-connected outerplane bipartite graph. Assume that s is a reducible face of G. Then s is adjacent to exactly one finite face of G. Proof. The boundary of a reducible face s of G is an even cycle and can be denoted by ∂ s = u1 u2 · · · u2p u1 where p ≥ 2. Recall that s is a peripheral face of G and the common periphery of s and G is an odd length handle P [1]. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then H is elementary by the definition of a reducible face of G. Suppose that s is adjacent to more than one finite face of G. Then ∂ s has more than two edges that are not on P since ∂ s is an even cycle and P is an odd length path. Without loss of generality, we can assume that P is a single edge, that is, P = e1 = u1 u2 , and no multiple edges can be created by doing so. Let C be the boundary of the infinite face of G. Then C is a Hamiltonian cycle of G. Note that ∂ s has at least three edges that are not peripheral edges of G, and so these edges are chords of C . We can denote these edges ei = ui ui+1 for 2 ≤ i < 2p and e2p = u2p u1 where p ≥ 2. Moreover, each of these edges ei (2 ≤ i ≤ 2p) partition the cycle C into two new cycles, the one whose interior has empty intersection with the interior of s is denoted by C (ei ). Let Aj (2 ≤ j ≤ p) be the subgraph of G that contains all vertices within and on the cycle C (e2j−1 ), and Bj (1 ≤ j ≤ p) be the subgraph of G that contains all vertices within and on the cycle C (e2j ). Then Aj (2 ≤ j ≤ p) and Bj (1 ≤ j ≤ p) are 2-connected outerplane bipartite graphs. We can see that G contains at least three subgraphs B1 , A2 , and B2 since p ≥ 2. See Fig. 2. Let A∗j = Aj − {u2j−1 , u2j } for 2 ≤ j ≤ p. Assume that M is a perfect matching of G. Let M |A∗ (2 ≤ j ≤ p) be the restriction of j M on A∗j . Let M |Bj (1 ≤ j ≤ p) be the restriction of M on Bj . Let M∗ (G; e1 ) be the set of perfect matchings of G not containing the edge P = e1 = u1 u2 . By Lemma 3 in [9], any M ∈ M∗ (G; e1 ) can be expressed as p
p
M = (∪j=2 M |A∗ ) ∪ (∪j=1 M |Bj ). j
Note that any perfect matching of H can be extended to a perfect matching M of G such that M ∈ M∗ (G; e1 ). It follows that any perfect matching of H cannot contain edges u2j−1 u2j of Aj for all 2 ≤ j ≤ p. On the other hand, each edge of H is contained a perfect matching of H since H is an elementary bipartite graph. This is a contradiction. Therefore, s is adjacent to exactly one finite face of G. □ The following theorem generalizes the results of [3] and [4] for the cases when G is a catacondensed benzenoid graph or a catacondensed even ring system. Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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Fig. 2. A 2-connected outerplane bipartite graph with a peripheral face s.
Theorem 3.2. Let G be a 2-connected outerplane bipartite graph. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then R(G) can be obtained from R(H) by a peripheral convex expansion, that is, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class of R(G) with the face-label s. Moreover, (i) R(G) has exactly one more Θ -class than R(H) and it has the face-label s, and (ii) each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label (adding more edges if needed). Proof. By Proposition 3.1, s is adjacent to exactly one finite face of G. Then similarly to the argument of Corollary 4.9 in [1], we can show that s is M-resonant for any perfect matching M in M(G; P + ). By Theorem 2.1, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class in R(G) with the face-label s. Note that all edges of R(G) that are not in R(H) are properly contained in a Cartesian product T □K2 by the structure of a peripheral convex expansion. Furthermore, any two antipodal edges of an induced 4-cycle are in relation Θ by the definition. It follows that R(G) has exactly one more Θ -class than R(H) and it has the face-label s. Each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label as follows: A Θ -class of R(G) has the same edges as the corresponding Θ -class of R(H) if the latter does not contain any edge of T . Otherwise, a Θ -class of R(G) can be obtained by doubling edges contained in T of the corresponding Θ -class of R(H). □ Corollary 3.3. Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. Assume that G has a RFD(G1 , G2 . . . , Gn (= G)) associated with a sequence of finite faces si (1 ≤ i ≤ n) and a sequence of odd length ears Pi (1 ≤ i ≤ n). Then R(G) can be obtained from the one edge graph by a sequence of peripheral convex expansions. Furthermore, R(G1 ) = K2 where the edge has the face-label s1 ; for 2 ≤ i ≤ n, R(Gi ) = pce(R(Gi−1 ); Ti−1 ) where the set of all edges between R(Gi−1 ) and Ti−1 is a Θ -class in R(Gi ) with the face-label si , R(Gi ) has exactly one more Θ -class than R(Gi−1 ) and it has the face-label si , each of other Θ -classes of R(Gi ) can be obtained from the corresponding Θ -class of R(Gi−1 ) with the same face-label (adding more edges if needed). Proof. It is trivial that R(G1 ) = K2 where the edge has the face-label s1 . For 2 ≤ i ≤ n, the finite face si is a reducible face of Gi , the ear Pi is an odd length handle of Gi that is the common periphery of si and Gi , and Gi−1 is the subgraph of Gi which can be obtained by removing all internal vertices (if exist) and edges of Pi . The result follows immediately by Theorem 3.2. □ By Corollary 3.3, if G is a 2-connected outerplane bipartite graph, then the set of Θ -classes of R(G) is 1–1 corresponding to the set of finite faces of G. We use sΘ to denote the Θ -class of R(G) with the face-label s. Theorem 3.4. Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. Then the graph Θ (R(G)) induced by the Θ -classes of R(G) is a tree and isomorphic to the inner dual of G. Proof. Induction on the number of finite faces of G. If G has exactly one finite face, then R(G) is an edge and the inner dual of G is a single vertex graph. So, Θ (R(G)) is a single vertex graph which is isomorphic to the inner dual of G. Assume that it is true when G has n − 1 finite faces. Let G be a 2-connected outerplane bipartite graph with n(> 1) finite faces. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then H is a 2-connected outerplane bipartite graph with n − 1 finite faces. By the induction hypothesis, Θ (R(H)) is a tree and isomorphic to the inner dual of H. By Proposition 3.1, s is adjacent to exactly one finite face s′ of G. Hence, the inner dual of G can be obtained from the inner dual of H by adding one pendent edge. By Theorem 3.2, R(G) can be obtained from R(H) by a peripheral convex expansion, that is, R(G) = pce(R(H); T ) where the set Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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of all edges between R(H) and T is a Θ -class in R(G) with the face-label s. Moreover, R(G) has exactly one more Θ -class than R(H) and it has the face-label s, and each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label (adding more edges if needed). Therefore, Θ (R(G)) has exactly one more vertex sΘ than Θ (R(H)). It remains to show that the vertex sΘ is adjacent to exactly one vertex s′Θ in Θ (R(G)). By Proposition 3.1 in [1], face-labels of two incident edges of a 4-cycle in R(G) are vertex-disjoint faces of G. Note that if two edges of R(G) with face-labels s1 and s2 are incident to a vertex M in R(G), then M is a perfect matching of G and both finite faces s1 and s2 are M-resonant faces of G. It follows that if s1 and s2 have a common vertex, then s1 and s2 must have a common edge and so they are adjacent in G. By the definition of the adjacency of two Θ -classes of R(G), we can see that the vertex sΘ is adjacent to exactly one vertex s′Θ in Θ (R(G)) since s is adjacent to exactly one finite face s′ in G. Therefore, Θ (R(G)) can be obtained from Θ (R(H)) by adding one pendent edge. It follows that Θ (R(G)) is a tree and isomorphic to the inner dual of G. □ Acknowledgments The author would like to thank the referees for their helpful comments. References [1] Z. Che, Structural properties of resonance graphs of plane elementary bipartite graphs, Discrete Appl. Math. 247 (2018) 102–110. [2] R. Hammack, W. Imrich, S. Klavžar, Handbook of product graphs, in: Discrete Mathematics and its Applications (Boca Raton), Second ed., CRC Press, Boca Raton, FL, 2011. [3] S. Klavžar, A. Vesel, P. Žigert, On resonance graphs of catacondensed hexagonal graphs: Structure, coding, and Hamiltonian path algorithm, MATCH Commun. Math. Comput. Chem. 49 (2003) 99–116. [4] S. Klavžar, P. Žigert, G. Brinkmann, Resonance graphs of catacondensed even ring systems are median, Discrete Math. 253 (2002) 35–43. [5] L. Lovasz, M.D. Plummer, Matching theory, in: North-Holland Mathematics Studies, 121, in: Annals of Discrete Mathematics, vol. 29, North-Holland Publishing Co., Amsterdam, 1986. [6] A. Vesel, Characterization of resonance graphs of catacondensed hexagonal graphs, MATCH Commun. Math. Comput. Chem. 53 (2005) 195–208. [7] H. Zhang, P.C.B. Lam, W.C. Shiu, Resonance graphs and a binary coding for the 1-factors of benzenoid systems, SIAM J. Discrete Math. 22 (2008) 971–984. [8] H. Zhang, F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291–311. [9] H. Zhang, L. Zhao, H. Yao, The Z-transformation graph for an outerplane bipartite graph has a Hamilton path, Appl. Math. Lett. 17 (2004) 897–901.
Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
Contents lists available at ScienceDirect
Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
Note
A characterization of the resonance graph of an outerplane bipartite graph Zhongyuan Che Department of Mathematics, Penn State University, Beaver Campus, Monaca, PA 15061, USA
article
info
Article history: Received 6 June 2018 Received in revised form 24 October 2018 Accepted 26 November 2018 Available online xxxx Keywords: Djoković–Winkler relation Θ Median graph Outerplane bipartite graph Peripheral convex expansion Reducible face Resonance graph Z -transformation graph
a b s t r a c t Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. It is known that R(G) is a median graph. Assume that s is a reducible face of G and H is the subgraph of G obtained by removing all internal vertices (if exist) and edges on the common periphery of s and G. We show that R(G) can be obtained from R(H) by a peripheral convex expansion. As an application, we prove that Θ (R(G)) is a tree and isomorphic to the inner dual of G, where Θ (R(G)) is the induced graph on the Djoković–Winkler relation Θ -classes of R(G). © 2018 Elsevier B.V. All rights reserved.
1. Introduction Let G be a plane bipartite graph with a perfect matching. The resonance graph of G, denoted by R(G), is the graph whose vertices are the perfect matchings of G, and two vertices M1 and M2 of R(G) are adjacent if and only if their symmetric difference is the periphery of a finite face s of G, and we say that the edge M1 M2 has the face-label s. The resonance graph of G is also called the Z -transformation graph of G. See Fig. 1 for the resonance graph R(G) of a 2-connected outerplane bipartite graph G. A peripheral face s of a plane elementary bipartite graph G is called reducible if the subgraph H of G obtained by removing all internal vertices (if exist) and edges on the common periphery of s and G is elementary. We gave a necessary and sufficient condition on when R(G) can be obtained from R(H) by a peripheral convex expansion with respect to a reducible face s of G in [1]. A peripheral expansion structure for the resonance graph of a catacondensed benzenoid graph (also called catacondensed hexagonal graph) was given in [3], and a peripheral convex expansion structure for the resonance graph of a catacondensed even ring system was given in [4]. A characterization of the resonance graph of a catacondensed hexagonal graph was presented in [6] in terms of the induced graph on the Djoković–Winkler relation Θ -classes of its resonance graph. In this paper, let G be a 2-connected outerplane bipartite graph. We first show that if s is a reducible face of G, then s has common edge(s) with exactly one finite face of G. By the decomposition structure of R(G) from [1], it follows that R(G) can be obtained from R(H) by a peripheral convex expansion. We conclude the paper with a characterization of R(G) using the tool of Djoković–Winkler relation Θ : The induced graph Θ (R(G)) on the Θ -classes of R(G) is a tree and isomorphic to the inner dual of G. This generalizes the corresponding results given in [3,4] and [6]. E-mail address: [email protected]. https://doi.org/10.1016/j.dam.2018.11.032 0166-218X/© 2018 Elsevier B.V. All rights reserved.
Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
2
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Fig. 1. A 2-connected outerplane bipartite graph G with a perfect matching M ′ . R(G) is the resonance graph of G and Θ (R(G)) is the induced graph on
Θ -classes of R(G).
2. Preliminaries A perfect matching (or, 1-factor) of a graph is a set of pairwise disjoint edges of the graph that covers all its vertices. Let M be a perfect matching of a graph G. An M-alternating cycle (resp., M-alternating path) is a cycle (resp., path) of G whose edges are alternatively in and off M. A plane graph is a graph in the plane where any two edges are either disjoint or meet only at a common end vertex. Each interior region of a plane graph G is called a finite face (or, an inner face) of G, and the exterior region of G is called the infinite face (or, exterior face) of G. Two faces (one can be the infinite face) of a plane graph G are adjacent if their peripheries have common edges. A finite face s of G is called a peripheral face if it is adjacent to the infinite face of G. The inner dual of a plane graph G is a plane graph whose vertex set is in 1–1 correspondence with the set of finite faces of G, and two vertices are adjacent if the corresponding finite faces of G are adjacent. If a plane graph G is 2-connected, then the periphery of any face of G is a cycle. The periphery of a finite face s of G is denoted by ∂ s. The periphery of the infinite face of G is denoted by ∂ G, which is referred as the periphery (or, boundary) of G. A face (including the infinite face) of G is called M-resonant if its periphery is an M-alternating cycle for a perfect matching M of G. A vertex of a plane graph is called an exterior vertex if it is located on the periphery of the graph, and an interior vertex otherwise. An even ring system is a 2-connected plane bipartite graph whose interior vertices are degree-3 vertices and exterior vertices are degree-2 or degree-3 vertices. An outerplane graph is a plane graph whose vertices are all exterior vertices. A catacondensed even ring system is an outerplane graph that is also an even ring system. A catacondensed benzenoid graph is a catacondensed even ring system whose finite faces are hexagons. A bipartite graph G is elementary if and only if it is connected and each edge is contained in some perfect matching of G [5]. We assume that all vertices of a bipartite graph are colored white and black such that adjacent vertices have different colors. A bipartite graph G is elementary if and only if it has a bipartite ear decomposition G = e + P1 + P2 + · · · + Pn starting from an edge G0 = e, and Gi is obtained from Gi−1 = e + P1 +· · ·+ Pi−1 by adding the ith ear Pi of odd length such that Pi joins two vertices in different colors of Gi−1 and Pi has no internal vertices in common with the vertices of Gi−1 [5]. A bipartite ear decomposition of a plane elementary bipartite graph G is called a reducible face decomposition (briefly, RFD) [8] if G1 is a boundary of a finite face s1 of G, and the ith ear Pi lies in the exterior of Gi−1 such that Pi and a part of the periphery of Gi−1 surround a finite face si of G for all 2 ≤ i ≤ n. For any RFD(G1 , G2 , . . . , Gn (= G)) of G associated with the face sequence si (1 ≤ i ≤ n), each si is a reducible face of Gi for 2 ≤ i ≤ n. Note that any 2-connected outerplane bipartite graph is elementary and so has a reducible face decomposition. For example, the 2-connected outerplane bipartite graph G in Fig. 1 has a RFD(G0 , G1 , . . . , G6 (= G)) associated with the face sequence si (0 ≤ i ≤ 6). The interval between two vertices u and v of a graph G is the set of all vertices on shortest paths between u and v in G, and denoted by IG (u, v ). A connected graph G is called a median graph if IG (u, v ) ∩ IG (u, w ) ∩ IG (v, w ) is a unique vertex for every triple of its vertices u, v, w . The Djoković–Winkler relation Θ plays an important role on the structural characterization of median graphs. Two edges uv and xy of a graph G are said to be in relation Θ , denoted by uv Θ xy, if dG (u, x) + dG (v, y) ̸ = dG (u, y) + dG (v, x). If G is a median graph, then the relation Θ defines an equivalence relation on the edge set of G [2]. Let G be a plane elementary bipartite graph and R(G) be its resonance graph. Then R(G) is a median graph [7]. Let xy be an edge of R(G) and Fxy = {e ∈ E(R(G)) | eΘ xy} be the set of all edges in relation Θ with xy in R(G). By Proposition 3.2 in [1], all edges in Fxy have the same face-label. On the other hand, two edges with the same face-label can be in different Θ -classes of R(G). The induced graph Θ (R(G)) on the Θ -classes of R(G) is a graph whose vertex set is the set of Θ -classes, and two vertices E and F of Θ (R(G)) are adjacent if R(G) has two incident edges e ∈ E and f ∈ F such that e and f are not contained in a common 4-cycle of R(G). See Fig. 1. Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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Let M be a perfect matching of a graph G. A path P of G is weakly M-augmenting [1] if it satisfies one of the following conditions: (i) P has length 1 and the single edge of P is not contained in M, (ii) P is an M-alternating path such that its two end edges are not contained in M. A quick remark is that a weakly M-augmenting path is different from an M-augmenting path defined in [5], where the path has length > 1 and its two end vertices are not covered by a non-perfect matching M. (There is a typo in [1] for the remark where ‘‘non-’’ was missed.) A path in a graph is called a handle if both its two end vertices have degree > 2 but all its internal vertices (if any) are degree-2 vertices of the graph. Let s be a reducible face of a plane elementary bipartite graph G. Then the common periphery of s and G is an odd length handle P [1]. By Proposition 4.1 in [1], the set of all perfect matchings M(G) of G has a partition M(G) = M(G; P − ) ∪ M(G; P + ), where M(G; P − ) is the set of perfect matchings M of G such that P is weakly M-augmenting, M(G; P + ) is the set of perfect matchings M of G such that P is not weakly M-augmenting. Let G be a graph with the vertex set V (G). An induced subgraph of G generated by a subset W ⊆ V (G) is denoted by ⟨W ⟩. An induced subgraph ⟨W ⟩ of G is called a convex subgraph if all shortest paths between u and v are contained in ⟨W ⟩ for any u, v ∈ W . Let T be a convex subgraph of G such that V (T ) ⊂ V (G). The peripheral convex expansion of G with respect to T , denoted by pce(G, T ), is the graph obtained from G by the following procedure [2]. 1. Replace each vertex v of T by an edge v1 v2 . 2. Insert edges between v1 and the neighbors of v in V (G) \ V (T ). 3. Insert the edges u1 v1 and u2 v2 whenever u, v of T are adjacent in G. A periphery expansion pe(G; T ) where T is just a subgraph of G can be obtained as the same procedure for pce(G; T ) by dropping the convex condition on T . A periphery expansion structure on R(G) was given in [3] when G is a catacondensed benzenoid graph, and a periphery convex expansion structure on R(G) was given in [4] when G is a catacondensed even ring system. We provided a necessary and sufficient condition on when R(G) can be obtained by a peripheral convex expansion with respect to a reducible face s of G in [1]. Theorem 2.1 ([1]). Let G be a plane elementary bipartite graph and R(G) be its resonance graph. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then R(G) can be obtained from R(H) by a peripheral convex expansion if and only if s is M-resonant for any perfect matching M in M(G; P + ). Moreover, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class of R(G) with the face-label s. 3. Main results Proposition 3.1. Let G be a 2-connected outerplane bipartite graph. Assume that s is a reducible face of G. Then s is adjacent to exactly one finite face of G. Proof. The boundary of a reducible face s of G is an even cycle and can be denoted by ∂ s = u1 u2 · · · u2p u1 where p ≥ 2. Recall that s is a peripheral face of G and the common periphery of s and G is an odd length handle P [1]. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then H is elementary by the definition of a reducible face of G. Suppose that s is adjacent to more than one finite face of G. Then ∂ s has more than two edges that are not on P since ∂ s is an even cycle and P is an odd length path. Without loss of generality, we can assume that P is a single edge, that is, P = e1 = u1 u2 , and no multiple edges can be created by doing so. Let C be the boundary of the infinite face of G. Then C is a Hamiltonian cycle of G. Note that ∂ s has at least three edges that are not peripheral edges of G, and so these edges are chords of C . We can denote these edges ei = ui ui+1 for 2 ≤ i < 2p and e2p = u2p u1 where p ≥ 2. Moreover, each of these edges ei (2 ≤ i ≤ 2p) partition the cycle C into two new cycles, the one whose interior has empty intersection with the interior of s is denoted by C (ei ). Let Aj (2 ≤ j ≤ p) be the subgraph of G that contains all vertices within and on the cycle C (e2j−1 ), and Bj (1 ≤ j ≤ p) be the subgraph of G that contains all vertices within and on the cycle C (e2j ). Then Aj (2 ≤ j ≤ p) and Bj (1 ≤ j ≤ p) are 2-connected outerplane bipartite graphs. We can see that G contains at least three subgraphs B1 , A2 , and B2 since p ≥ 2. See Fig. 2. Let A∗j = Aj − {u2j−1 , u2j } for 2 ≤ j ≤ p. Assume that M is a perfect matching of G. Let M |A∗ (2 ≤ j ≤ p) be the restriction of j M on A∗j . Let M |Bj (1 ≤ j ≤ p) be the restriction of M on Bj . Let M∗ (G; e1 ) be the set of perfect matchings of G not containing the edge P = e1 = u1 u2 . By Lemma 3 in [9], any M ∈ M∗ (G; e1 ) can be expressed as p
p
M = (∪j=2 M |A∗ ) ∪ (∪j=1 M |Bj ). j
Note that any perfect matching of H can be extended to a perfect matching M of G such that M ∈ M∗ (G; e1 ). It follows that any perfect matching of H cannot contain edges u2j−1 u2j of Aj for all 2 ≤ j ≤ p. On the other hand, each edge of H is contained a perfect matching of H since H is an elementary bipartite graph. This is a contradiction. Therefore, s is adjacent to exactly one finite face of G. □ The following theorem generalizes the results of [3] and [4] for the cases when G is a catacondensed benzenoid graph or a catacondensed even ring system. Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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Fig. 2. A 2-connected outerplane bipartite graph with a peripheral face s.
Theorem 3.2. Let G be a 2-connected outerplane bipartite graph. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then R(G) can be obtained from R(H) by a peripheral convex expansion, that is, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class of R(G) with the face-label s. Moreover, (i) R(G) has exactly one more Θ -class than R(H) and it has the face-label s, and (ii) each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label (adding more edges if needed). Proof. By Proposition 3.1, s is adjacent to exactly one finite face of G. Then similarly to the argument of Corollary 4.9 in [1], we can show that s is M-resonant for any perfect matching M in M(G; P + ). By Theorem 2.1, R(G) = pce(R(H); T ) where the set of all edges between R(H) and T is a Θ -class in R(G) with the face-label s. Note that all edges of R(G) that are not in R(H) are properly contained in a Cartesian product T □K2 by the structure of a peripheral convex expansion. Furthermore, any two antipodal edges of an induced 4-cycle are in relation Θ by the definition. It follows that R(G) has exactly one more Θ -class than R(H) and it has the face-label s. Each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label as follows: A Θ -class of R(G) has the same edges as the corresponding Θ -class of R(H) if the latter does not contain any edge of T . Otherwise, a Θ -class of R(G) can be obtained by doubling edges contained in T of the corresponding Θ -class of R(H). □ Corollary 3.3. Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. Assume that G has a RFD(G1 , G2 . . . , Gn (= G)) associated with a sequence of finite faces si (1 ≤ i ≤ n) and a sequence of odd length ears Pi (1 ≤ i ≤ n). Then R(G) can be obtained from the one edge graph by a sequence of peripheral convex expansions. Furthermore, R(G1 ) = K2 where the edge has the face-label s1 ; for 2 ≤ i ≤ n, R(Gi ) = pce(R(Gi−1 ); Ti−1 ) where the set of all edges between R(Gi−1 ) and Ti−1 is a Θ -class in R(Gi ) with the face-label si , R(Gi ) has exactly one more Θ -class than R(Gi−1 ) and it has the face-label si , each of other Θ -classes of R(Gi ) can be obtained from the corresponding Θ -class of R(Gi−1 ) with the same face-label (adding more edges if needed). Proof. It is trivial that R(G1 ) = K2 where the edge has the face-label s1 . For 2 ≤ i ≤ n, the finite face si is a reducible face of Gi , the ear Pi is an odd length handle of Gi that is the common periphery of si and Gi , and Gi−1 is the subgraph of Gi which can be obtained by removing all internal vertices (if exist) and edges of Pi . The result follows immediately by Theorem 3.2. □ By Corollary 3.3, if G is a 2-connected outerplane bipartite graph, then the set of Θ -classes of R(G) is 1–1 corresponding to the set of finite faces of G. We use sΘ to denote the Θ -class of R(G) with the face-label s. Theorem 3.4. Let G be a 2-connected outerplane bipartite graph and R(G) be its resonance graph. Then the graph Θ (R(G)) induced by the Θ -classes of R(G) is a tree and isomorphic to the inner dual of G. Proof. Induction on the number of finite faces of G. If G has exactly one finite face, then R(G) is an edge and the inner dual of G is a single vertex graph. So, Θ (R(G)) is a single vertex graph which is isomorphic to the inner dual of G. Assume that it is true when G has n − 1 finite faces. Let G be a 2-connected outerplane bipartite graph with n(> 1) finite faces. Assume that s is a reducible face of G and P is the common periphery of s and G. Let H be the subgraph of G obtained by removing all internal vertices (if exist) and edges of P. Then H is a 2-connected outerplane bipartite graph with n − 1 finite faces. By the induction hypothesis, Θ (R(H)) is a tree and isomorphic to the inner dual of H. By Proposition 3.1, s is adjacent to exactly one finite face s′ of G. Hence, the inner dual of G can be obtained from the inner dual of H by adding one pendent edge. By Theorem 3.2, R(G) can be obtained from R(H) by a peripheral convex expansion, that is, R(G) = pce(R(H); T ) where the set Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.
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of all edges between R(H) and T is a Θ -class in R(G) with the face-label s. Moreover, R(G) has exactly one more Θ -class than R(H) and it has the face-label s, and each of other Θ -classes of R(G) can be obtained from the corresponding Θ -class of R(H) with the same face-label (adding more edges if needed). Therefore, Θ (R(G)) has exactly one more vertex sΘ than Θ (R(H)). It remains to show that the vertex sΘ is adjacent to exactly one vertex s′Θ in Θ (R(G)). By Proposition 3.1 in [1], face-labels of two incident edges of a 4-cycle in R(G) are vertex-disjoint faces of G. Note that if two edges of R(G) with face-labels s1 and s2 are incident to a vertex M in R(G), then M is a perfect matching of G and both finite faces s1 and s2 are M-resonant faces of G. It follows that if s1 and s2 have a common vertex, then s1 and s2 must have a common edge and so they are adjacent in G. By the definition of the adjacency of two Θ -classes of R(G), we can see that the vertex sΘ is adjacent to exactly one vertex s′Θ in Θ (R(G)) since s is adjacent to exactly one finite face s′ in G. Therefore, Θ (R(G)) can be obtained from Θ (R(H)) by adding one pendent edge. It follows that Θ (R(G)) is a tree and isomorphic to the inner dual of G. □ Acknowledgments The author would like to thank the referees for their helpful comments. References [1] Z. Che, Structural properties of resonance graphs of plane elementary bipartite graphs, Discrete Appl. Math. 247 (2018) 102–110. [2] R. Hammack, W. Imrich, S. Klavžar, Handbook of product graphs, in: Discrete Mathematics and its Applications (Boca Raton), Second ed., CRC Press, Boca Raton, FL, 2011. [3] S. Klavžar, A. Vesel, P. Žigert, On resonance graphs of catacondensed hexagonal graphs: Structure, coding, and Hamiltonian path algorithm, MATCH Commun. Math. Comput. Chem. 49 (2003) 99–116. [4] S. Klavžar, P. Žigert, G. Brinkmann, Resonance graphs of catacondensed even ring systems are median, Discrete Math. 253 (2002) 35–43. [5] L. Lovasz, M.D. Plummer, Matching theory, in: North-Holland Mathematics Studies, 121, in: Annals of Discrete Mathematics, vol. 29, North-Holland Publishing Co., Amsterdam, 1986. [6] A. Vesel, Characterization of resonance graphs of catacondensed hexagonal graphs, MATCH Commun. Math. Comput. Chem. 53 (2005) 195–208. [7] H. Zhang, P.C.B. Lam, W.C. Shiu, Resonance graphs and a binary coding for the 1-factors of benzenoid systems, SIAM J. Discrete Math. 22 (2008) 971–984. [8] H. Zhang, F. Zhang, Plane elementary bipartite graphs, Discrete Appl. Math. 105 (2000) 291–311. [9] H. Zhang, L. Zhao, H. Yao, The Z-transformation graph for an outerplane bipartite graph has a Hamilton path, Appl. Math. Lett. 17 (2004) 897–901.
Please cite this article as: Z. Che, A characterization of the resonance graph of an outerplane bipartite graph, Discrete Applied Mathematics (2018), https://doi.org/10.1016/j.dam.2018.11.032.