Diameter-Preserving Spanning Trees in Sparse Weighted Graphs

Abstract

We give a simple sufficient condition for a weighted graph to have a diameter-preserving spanning tree. More precisely, let G = (V, E, f E ) be a connected edge weighted graph with f E being the edge weight function. Let f V be the vertex weight function of G induced by f E as follows: f V (v) = max{f E (e) : e is incident with v} for all $${v \in V}$$. We show that G contains a diameter-preserving spanning tree if $${d(G)\ge \frac{2}{3} \sum_{v\in V} f_V(v)}$$where d(G) is the diameter of G. The condition is sharp in the sense that for any $${\epsilon >0 }$$, there exist weighted graphs G satisfying $${d(G) > (\frac{2}{3}-\epsilon)\sum_{v\in V} f_V(v)}$$and not containing a diameter-preserving spanning tree.