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    Beyond Helly graphs: the diameter problem on absolute retracts

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    Characterizing the graph classes such that, on nn-vertex mm-edge graphs in the class, we can compute the diameter faster than in O(nm){\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 HH of a graph GG is called a retract of GG if it is the image of some idempotent endomorphism of GG. Two necessary conditions for HH being a retract of GG is to have HH is an isometric and isochromatic subgraph of GG. We say that HH is an absolute retract of some graph class C{\cal C} if it is a retract of any G∈CG \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 O~(mn)\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 kk-chromatic graphs, for every fixed k≥3k \geq 3. Finally, we study the diameter problem within the absolute retracts of planar graphs and split graphs, respectively
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