Obstructions to faster diameter computation: Asteroidal sets
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An extremity is a vertex such that the removal of its closed neighbourhood does not increase the number of connected components. Let Ext_α be the class of all connected graphs whose quotient graph obtained from modular decomposition contains no more than α pairwise nonadjacent extremities. Our main contributions are as follows. First, we prove that the diameter of every m-edge graph in Ext_α can be computed in deterministic O(α^3 m^3/2) time. We then improve the runtime to linear for all graphs with bounded clique-number. Furthermore, we can compute an additive +1-approximation of all vertex eccentricities in deterministic O(α^2 m) time. This is in sharp contrast with general m-edge graphs for which, under the Strong Exponential Time Hypothesis (SETH), one cannot compute the diameter in O(m^2-ϵ) time for any ϵ > 0. As important special cases of our main result, we derive an O(m^3/2)-time algorithm for exact diameter computation within dominating pair graphs of diameter at least six, and an O(k^3m^3/2)-time algorithm for this problem on graphs of asteroidal number at most k. We end up presenting an improved algorithm for chordal graphs of bounded asteroidal number, and a partial extension of our results to the larger class of all graphs with a dominating target of bounded cardinality. Our time upper bounds in the paper are shown to be essentially optimal under plausible complexity assumptions.
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