On graphs coverable by chubby shortest paths

TL;DR

The paper extends the geodesic-cover framework to a coarse setting, showing that if a graph $G$ admits $k$ geodesic shortest paths within distance $\rho$ of every vertex, then $G$ is $(3,12\rho)$-quasi-isometric to a graph with pathwidth $k^{\mathcal{O}(k)}$ and maximum degree $\mathcal{O}(k)$. It constructs a distance-$2\rho$ path-partition-decomposition whose bags are coverable by $k^{\mathcal{O}(k)}$ balls, and demonstrates a practical quasi-isometry via the distance-$4\rho$ independent-set graph $H_{4\rho}(I,G)$ and its subdivision. The authors develop the conceptual machinery of snapwalks and simplified variants to relate shortest paths to geodesic covers, enabling layer-wise coverability arguments and DP-based algorithmic results. Consequently, they obtain XP-time algorithms for computing distance-based packing and domination parameters, and provide a clearer coarse-structural understanding that improves prior bounds from $2^{\mathcal{O}(k)}$ to $k^{\mathcal{O}(k)}$. The work has potential impact on algorithmic graph problems in networks where metric structure is governed by a small set of geodesics.

Abstract

Dumas, Foucaud, Perez, and Todinca [SIAM J. Disc. Math., 2024] proved that if the vertex set of a graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is bounded by $\mathcal{O}(k \cdot 3^k)$. We prove a coarse variant of this theorem: if in a graph $G$ one can find~$k$ shortest paths such that every vertex is at distance at most $ρ$ from one of them, then $G$ is $(3,12ρ)$-quasi-isometric to a graph of pathwidth $k^{\mathcal{O}(k)}$ and maximum degree $\mathcal{O}(k)$, and $G$ admits a path-partition-decomposition whose bags are coverable by $k^{\mathcal{O}(k)}$ balls of radius at most $2ρ$ and vertices from non-adjacent bags are at distance larger than $2ρ$. We also discuss applications of such decompositions in the context of algorithms for finding maximum distance independent sets and minimum distance dominating sets in graphs.

Figures (3)

• Figure 1: Illustration of case $\textbf{(1)}.$
• Figure 2: Illustration of case $\textbf{(2)}.$
• Figure 3: Illustration of case $\textbf{(3)}.$

Theorems & Definitions (30)

• Theorem 1.1: dumas2024geodesics
• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Corollary 1.2
• Lemma 2.0
• proof
• Lemma 2.0: abrishami2025coarsetreedecompositionscoarse
• Lemma 3.1
• proof