On graphs coverable by k shortest paths

TL;DR

This work proves that if a graph's edges (or vertices) can be covered by at most $k$ shortest paths, then its pathwidth is bounded by a function of $k$ (specifically $pw(G)=O(3^k)$ for edges and $pw(G)=O(k\cdot3^k)$ for vertices). These structural bounds enable MSOL-based (Courcelle) techniques to solve Isometric Path Cover with Terminals and Strong Geodetic Set with Terminals in FPT time when parameterized by the number of terminals, while the non-terminal variants lie in XP. The core contributions include a factorial-to-single-exponential bound improvement via a $k$-labelled branching-tree construction and a careful colour-sign encoding for paths, which together connect covering properties to bounded pathwidth. The results have algorithmic significance, offering tractable routes for these distance-constrained covering problems on graphs with few terminals and highlighting open questions about tightening the bounds or extending to directed graphs. Overall, the paper advances the understanding of when distance-based path-cover problems become efficiently solvable on structurally restricted graphs.

Abstract

We show that if the edges or vertices of an undirected graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is upper-bounded by a single-exponential function of $k$. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph $G$ and a set of $k$ pairs of vertices called terminals, asks whether $G$ can be covered by $k$ shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph $G$ and a set of $k$ terminals, asks whether there exist $\binom{k}{2}$ shortest paths covering $G$, each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter $k$.

Figures (7)

• Figure 1: $(P_1,\operatorname{col}_1)$ and $(P_2,\operatorname{col}_2)$ are well-coloured shortest paths of same length. The vertex $x$ is shared by the two paths. One can replace $\mu_{\textcolor{blue}{blue}}[a,x]$ in $P_2$ by $\mu_{\textcolor{red}{red}}[a,x]$, this proves that $b=c$.
• Figure 2: Illustration of the configuration one obtains when $n$ is not the root, has at least two children $n_1$ and $n_2$ and parent $p$. Node $n'$ is an ancestor of $n$ while $n"$ is not.
• Figure 3: Example of the construction in the proof of \ref{['lem:boundak']} for a set of paths $\mathcal{P}_n$. In this example, all paths share the same subpath $(P_1[a,x_n],\operatorname{col})$, $\mathcal{C}_n =\{\textcolor{red}{red}, \textcolor{blue}{blue},\textcolor{mygreen}{green}\}$ and there are no edges with colour in $\mathcal{C}_n$ in the path $(P_i[w_{P_i}:],\operatorname{col}_i)$. The sets $\mathcal{P}_{(\textcolor{red}{red},-)}, \mathcal{P}_{(\textcolor{mygreen}{green},+)}, \mathcal{P}_{(\textcolor{mygreen}{green},-)}$ are empty.
• Figure 4: Two well-coloured paths $(P,\operatorname{col})$ and $(P',\operatorname{col}')$ with same colours-signs word $\omega=((red,-),(blue,+))$, same length $5$ and same start vertex $a$ but different end-vertices ($b$ and $c$).
• Figure 5: Example of the construction in proof of \ref{['lem:boundak_vertex']} for a set of paths $\mathcal{P}_n$. In this example, all paths share the same subpath $(P_1[a,x_n],\operatorname{col})$ except for $P_5$ that is subpath of the latter. Moreover these paths may have different length. We have $\mathcal{C}_n =\{\textcolor{red}{red}, \textcolor{blue}{blue},\textcolor{mygreen}{green}\}$ and there are no vertices with colour in $\mathcal{C}_n$ in the path $(P_i[w_{P_i}:],\operatorname{col}_i)$ except for $w_{P_i}$. The sets $\mathcal{P}_{(\textcolor{red}{red},-)},\mathcal{P}_{(\textcolor{mygreen}{green},+)}, \mathcal{P}_{(\textcolor{mygreen}{green},-)}$ are empty. The vertices $y_{P_1}$ and $y_{P_3} = y_{P_4}$ are the same if $x_n \notin V(\mu_{\textcolor{blue}{blue}})$ and distinct else. The path $P^*_2$ has the same length as $P_1$, the other modified paths might not have the same length than their counterpart, even $P^*_4$ as $z_{P_4}$ might be different from $y_{P_4}$. The path $P_5$ was arbitrarily added to $\mathcal{P}_{(\textcolor{blue}{blue},+)}$.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Lemma 1
• proof
• Lemma 2: Good colouring lemma
• proof
• Lemma 3
• proof