An eccentricity 2-approximating spanning tree of a chordal graph is computable in linear time

Abstract

It is known that every chordal graph G=(V,E) has a spanning tree T such that, for every vertex v∈V, eccT(v)≤eccG(v)+2 holds (here eccG(v):=max⁡{dG(v,u):u∈V} is the eccentricity of v in G). We show that such a spanning tree can be computed in linear time for every chordal graph. As a byproduct, we get that the eccentricities of all vertices of a chordal graph G can be computed in linear time with an additive one-sided error of at most 2, i.e., after a linear time preprocessing, for every vertex v of G, one can compute in O(1) time an estimate eˆ(v) of its eccentricity eccG(v) such that eccG(v)≤eˆ(v)≤eccG(v)+2.