A faster diameter problem algorithm for a chordal graph, with a connection to its center problem

Abstract

Let G be a non-trivial simple graph with vertices V(G)=V and edges E(G)=E, and let n=|V|,m=|E|. Computing the diameter of G and the min–max center of G (C(G)) are both quadratic-time (O(m2)). A problem is strongly subquadratic-time if it is O(m2−ϵ) for some ϵ>0. If either the diameter problem or the center problem is strongly subquadratic-time, then the Strong Exponential Time Hypothesis would be violated. The same is true even for chordal graphs (graphs having no induced n-cycle for n≥4). This paper presents an algorithm that is faster than existing algorithms for the diameter problem for all chordal graphs.With α(C(G)) the size of a largest independent vertex subset of C(G), it is proven here that the diameter problem for chordal graphs is O(α(C(G))m)-time. The algorithm does not require knowledge of C(G); nevertheless, it relates the diameter problem to the structure of C(G).Large-radius chordal graphs G are likely to have small α(C(G))|E(G)|, which suggests the existence of interesting classes of large-radius chordal graphs having strongly subquadratic-time diameter problems.