A polynomial bound on the pathwidth of graphs edge-coverable by $k$ shortest paths

TL;DR

This work strengthens the understanding of how edge-coverings by shortest paths constrain a graph's width parameters. The authors prove a polynomial bound $pw(G)=O(k^4)$ for graphs edge-coverable by $k$ shortest paths, improving the previous exponential guarantees, and they establish exact bounds $pw(G)\le k$ for $k\le 3$. A key technical contribution is a separator-based framework built around a $6k$-layered substructure that yields controlled pathwidth after removing a polynomial-size set. Additionally, they show that if $G$ is edge-coverable by two isometric trees, then $tw(G)\le 2$, contrasting with known vertex-cover phenomena. These results have potential algorithmic impact for Isometric Path Cover problems and related graph-routing questions, and they leave intriguing open questions about linear bounds and vertex-cover analogues.

Abstract

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one; namely, we show that every graph whose edge set can be covered by $k$ shortest paths has pathwidth $O(k^4)$, answering a question from the same paper. Moreover, we prove that when $k\leq 3$, every such graph has pathwidth at most $k$ (and this bound is tight). Finally, we show that even though there exist graphs with arbitrarily large treewidth whose vertex set can be covered by $2$ isometric trees, every graph whose set of edges can be covered by $2$ isometric trees has treewidth at most $2$.

Figures (18)

• Figure 1: This graph is vertex-coverable by $2$ isometric trees (one with red vertices, the other with blue vertices) and contains $K_{3,3}$ as a minor (black edges). Hence, it has treewidth at least $3$.
• Figure 2: The path $Q$ represented in green is reducing. $P_1$ is represented in blue.
• Figure 3: The red path represents some path $P\in \mathcal{P}_1$. The blue path is $P_1$.
• Figure 4: Configuration of \ref{['lem: keyP1']}.
• Figure 5: The path $L$ (in red) is not $\nearrow$-free.

Theorems & Definitions (72)

• Theorem 1.1: DFPT24
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Theorem 2.2: Kinnersley
• Lemma 2.3
• proof
• Theorem 3.1
• Theorem 3.1