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arXiv · 2502.07501

Faster diameter computation in graphs of bounded Euler genus

Kacper Kluk·Marcin Pilipczuk·Michał Pilipczuk·Giannos Stamoulis
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Abstract

We show that for any fixed integer $k \geq 0$, there exists an algorithm that computes the diameter and the eccentricies of all vertices of an input unweighted, undirected $n$-vertex graph of Euler genus at most $k$ in time \[ \mathcal{O}_k(n^{2-\frac{1}{25}}). \] Furthermore, for the more general class of graphs that can be constructed by clique-sums from graphs that are of Euler genus at most $k$ after deletion of at most $k$ vertices, we show an algorithm for the same task that achieves the running time bound \[ \mathcal{O}_k(n^{2-\frac{1}{356}} \log^{6k} n). \] Up to today, the only known subquadratic algorithms for computing the diameter in those graph classes are that of [Ducoffe, Habib, Viennot; SICOMP 2022], [Le, Wulff-Nilsen; SODA 2024], and [Duraj, Konieczny, Potępa; ESA 2024]. These algorithms work in the more general setting of $K_h$-minor-free graphs, but the running time bound is $\mathcal{O}_h(n^{2-c_h})$ for some constant $c_h > 0$ depending on $h$. That is, our savings in the exponent, as compared to the naive quadratic algorithm, are independent of the parameter $k$. The main technical ingredient of our work is an improved bound on the number of distance profiles, as defined in [Le, Wulff-Nilsen; SODA 2024], in graphs of bounded Euler genus.

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Kacper Kluk, Marcin Pilipczuk, Michał Pilipczuk, Giannos Stamoulis. 2025-02-11. Faster diameter computation in graphs of bounded Euler genus. https://arxiv.org/abs/2502.07501

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