Collective Tree 1-Spanners for Interval Graphs

Abstract

AbstractIn this paper we study the existence of a small set \(\mathcal{T}\) of spanning trees that collectively “1-span” an interval graph G. In particular, for any pair of vertices u,v we require a tree \(T \in \mathcal{T}\)such that the distance between u and v in T is at most one more than their distance in G. We show that:– there is no constant size set of collective tree 1-spanners for interval graphs (even unit interval graphs),– interval graph G has a set of collective tree 1-spanners of size O(log D), where D is the diameter of G,– interval graphs have a 1-spanner with fewer than 2n – 2 edges.Furthermore, at the end of the paper we state other results on collective tree c-spanners for c > 1 and other more general graph classes.KeywordsShort PathPlanar GraphInterval GraphChordal GraphShort Path TreeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.