Beyond Helly graphs: the diameter problem on absolute retracts

Abstract

International audienceA subgraph H of a graph G is called a retract of G if it is the image of some idempotent endomorphism of G. We say that H is an absolute retract of some graph class C if it is a retract of any G ∈ C of which it is an isochromatic and isometric subgraph. In this paper, we study the complexity of computing the diameter within the absolute retracts of various hereditary graph classes. First, we show how to compute the diameter within absolute retracts of bipartite graphs in randomized $\tilde{\cal O}(m √ n)$ time. Even on the proper subclass of cube-free modular graphs it is, to our best knowledge, the first subquadratic-time algorithm for diameter computation. For the special case of chordal bipartite graphs, it can be improved to linear time, and the algorithm even computes all the eccentricities. Then, we generalize these results to the absolute retracts of k-chromatic graphs, for every k ≥ 3. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs