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.