The biplanar tree graph

Abstract

The complete twisted graph of order n, denoted by \(T_n\), is a complete simple topological graph with vertices \(u_1, u_2, \ldots , u_n\) such that two edges \(u_iu_j\) and \(u_{i'}u_{j'}\) cross if and only if \( i<i'<j'<j \) or \( i'<i<j<j' \). The convex geometric complete graph of order n, denoted by \(G_n\), is a convex geometric graph with vertices \(v_1, v_2, \ldots , v_n\) placed counterclockwise, in which every pair of vertices is adjacent. A biplanar tree of order n is a labeled tree with vertex set \(\{v_1,v_2,\ldots , v_n \}\) having the property of being planar when embedded in both \(T_n\) and \(G_n\). Given a connected graph G the (combinatorial) tree graph \(\mathcal {T}(G)\) is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in \(\mathcal {T}(G)\) if there are edges \(e \in P\) and \(f\in Q\) such that \(Q=P - e + f\). For all positive integers n, we denote by \(\mathcal {T}(n)\) the graph \(\mathcal {T}(K_n)\). The biplanar tree graph, \(\mathcal {B}(n)\), is the subgraph of \(\mathcal {T}(n)\) induced by the biplanar trees of order n. In this paper we give a characterization of the biplanar trees and we study the structure, the radius and the diameter of the biplanar tree graph.

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Correspondence to Ana Paulina Figueroa.

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Partial support of Asociación de Cultura Mexicana A.C and CONACyT, México Projects A1-S-12891 and 47510664.

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Figueroa, A.P., Fresán-Figueroa, J. The biplanar tree graph. Bol. Soc. Mat. Mex. 26, 795–806 (2020). https://doi.org/10.1007/s40590-020-00287-y

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Mathematics Subject Classification

  • 05C10
  • 05C05
  • 05C76
  • 05C40