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 ;. . .; u n such that two edges u i u j and u i 0 u j 0 cross if and only if i\i 0 \j 0 \j or i 0 \i\j\j 0. The convex geometric complete graph of order n, denoted by G n , is a convex geometric graph with vertices v 1 ; v 2 ;. . .; 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 fv 1 ; v 2 ;. . .; v n g having the property of being planar when embedded in both T n and G n. Given a connected graph G the (combinatorial) tree graph T ðGÞ is the graph whose vertices are the spanning trees of G and two trees P and Q are adjacent in T ðGÞ if there are edges e 2 P and f 2 Q such that Q ¼ P À e þ f. For all positive integers n, we denote by T ðnÞ the graph T ðK n Þ. The biplanar tree graph, BðnÞ, is the subgraph of T ðnÞ 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.

Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

“…Originally introduced by Harborth and Mengersen in 1992 [ 21 ], twisted drawings have been thoroughly investigated; see for example [ 4 , 9 , 13 , 24 , 27 , 28 , 34 ]. Pach, Solymosi, and Tóth [ 28 ] showed that every simple drawing of contains an induced subdrawing of a complete graph on vertices which is either convex 3 or twisted .…”