Faster diameter computation in graphs of bounded Euler genus

TL;DR

The paper presents subquadratic algorithms for computing the diameter and all vertex eccentricities in graphs of bounded Euler genus and in graphs formed by clique-sums of such graphs after deleting a bounded number of vertices. The key technical advance is a polynomial bound, independent of the genus parameter, on the number of distance profiles, achieved by reducing to planar graphs via a careful decomposition along shortest paths and introducing anchor-distance profiles and hat-distances. This enables efficient batching and reuse of distance information through r-divisions and distance-profile data structures, extended to graphs with apices and to clique-sum constructions. The results substantively close the gap between subquadratic algorithms known for minor-free graphs and the more restricted genus-bounded setting, with implications for fast exact graph diameter computations in these structured graph classes and a robust framework for further generalizations.

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.

Figures (3)

• Figure 1: Illustration of a construction that shows that linear dependency on $h$ in the exponent of the bound on the number of profiles is inevitable, even in graphs of treewidth $h$.
• Figure 2: Left: A graph $G$ embedded on a surface $\Sigma$ and a subgraph $H$ of $G$ (in blue) that is a simple cut-graph of $\Sigma$. Right: The graph $G\textrm{\CutLeft} H$ embedded on the surface $\Sigma\textrm{\CutLeft} H$ (which is homeomorphic to a disk); the blue vertices/edges are copies of the vertices/edges of $H$.
• Figure 3: An illustration of (a part of) the construction of the graph $\widehat{G}$. The squared vertices are copies of anchor vertices. The marked squared vertex is also a copy of a vertex in $R$. The highlighted edges are copies of edges of $H$ in $G\textrm{\CutLeft} H$, while the paths obtained by subdividing the edges of $E_{\mathsf{out}}\cup E_{\mathsf{next}}$ are depicted with dashed edges. Edges adjacent to $t$ correspond to paths of appropriate length.

Theorems & Definitions (50)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3: LeW24
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: Sauer-Shelah Lemma
• Theorem 2.2
• Theorem 2.3: Willard85
• Theorem 2.4: WulffNilsen11