Beyond Helly graphs: the diameter problem on absolute retracts

TL;DR

This paper investigates the complexity of computing graph diameters within absolute retracts of various graph classes, providing efficient algorithms for bipartite, chordal bipartite, k-chromatic, planar, and split graphs.

Contribution

It introduces the concept of absolute retracts in graph classes and offers new algorithms with improved time complexities for diameter computation in these classes.

Findings

Linear time algorithm for chordal bipartite graphs

Randomized or bipartite graphs

Algorithms for k-chromatic, planar, and split graphs

Abstract

Characterizing the graph classes such that, on $n$-vertex $m$-edge graphs in the class, we can compute the diameter faster than in ${\cal O}(nm)$ time is an important research problem both in theory and in practice. We here make a new step in this direction, for some metrically defined graph classes. Specifically, a subgraph $H$ of a graph $G$ is called a retract of $G$ if it is the image of some idempotent endomorphism of $G$. Two necessary conditions for $H$ being a retract of $G$ is to have $H$ is an isometric and isochromatic subgraph of $G$. We say that $H$ is an absolute retract of some graph class ${\cal C}$ if it is a retract of any $G \in {\cal C}$ 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 $\tilde{\cal O}(m\sqrt{n})$ time. 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 fixed $k \geq 3$. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively.