Compatible spanning trees

Abstract

Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree of S. In this work, we study the problem of finding a second plane geometric tree T' spanning S, such that is compatible with T and shares the minimum number of edges with T. We prove that there is always a compatible plane geometric tree T' having at most #n - 3#/4 edges in common with T, and that for some plane geometric trees T, any plane tree T' spanning S, compatible with T, has at least #n - 2#/5 edges in common with T. #C# 2013 Elsevier B.V. All rights reserved.Preprin