DISTANCE PRESERVING SUBTREES IN MINIMUM AVERAGE DISTANCE SPANNING TREES

Abstract

Given an undirected graph G = (V, E) with n vertices and a positive length w(e) on each edge e ∈ E, we consider Minimum Average Distance (MAD) spanning trees i.e., trees that minimize the path length summed over all pairs of vertices. One of the first results on this problem is due to Wong who showed in 1980 that a Distance Preserving (DP) spanning tree rooted at the median of G is a 2-approximate solution. On the other hand, Dankelmann has exhibited in 2000 a class of graphs where no MAD spanning tree is distance preserving from a vertex. We establish here a new relation between MAD and DP trees in the particular case where the lengths are integers. We show that in a MAD spanning tree of G, each subtree H′ = (V′, E′) consisting of a vertex [Formula: see text] and the union of branches of [Formula: see text] that are each of size less than or equal to [Formula: see text], where w+ is the maximum edge-length in G, is a distance preserving spanning tree of the subgraph of G induced by V′.