Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number

TL;DR

The Geodetic Set problem asks whether a size-$k$ vertex subset S can cover all vertices by being on shortest paths between pairs in S. The authors show NP-hardness on graphs with constant pathwidth and feedback vertex set number by a reduction from $3$-Dimensional Matching, producing a graph $G$ with $\texttt{pw}(G) \le 17$ and $\texttt{fvs}(G) \le 13$, thereby answering whether an $\XP$ algorithm exists for the combined parameters in the negative. They introduce structured gadgets and a careful distance-based construction to encode the instance, and prove correctness via pendant-vertex coverage arguments and a witness set $Q$. This result highlights strong hardness for metric-graph problems under bounded-width parameters and clarifies the landscape between NP-hardness and algorithmic bounds for Geodetic Set, including connections to mixed-search-based width relaxations and treedepth-based approaches.

Abstract

In the \textsc{Geodetic Set} problem, the input consists of a graph $G$ and a positive integer $k$. The goal is to determine whether there exists a subset $S$ of vertices of size $k$ such that every vertex in the graph is included in a shortest path between two vertices in $S$. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is $\W[1]$-hard when parameterized by the pathwidth and the feedback vertex set number of the input graph. They posed the question of whether the problem admits an $\XP$ algorithm when parameterized by the combination of these two parameters. We answer this in negative by proving that the problem remains \NP-hard on graphs of constant pathwidth and feedback vertex set number.

Figures (2)

• Figure 1: The figure shows the $i^{th}$ copy of set-encoding gadget $\{z^i_2, z^i_1\} \cup X^i \cup Y^i$, element-encoding gadgets encoding elements of the form $(\alpha, a)$, $(\beta, b)$, $(\gamma, c)$ and $(\gamma, c')$, for some $a, b, c, c' \in [n]$, and the common branching vertices. Each solid (black or red) edge represents a path of fixed length which is either $M^2$ or $M^2 - 1$. Each dotted (blue or red) edge represents a path whose length depends on the set or element corresponding to one of its endpoints. Apart from the internal vertices in the red edges, all the other vertices are covered by pendant vertices in the graph. For clarity, we show connecting paths starting at only one vertex in $X^i$. Moreover, it only shows one of the nine vertices, viz $x_{p^{\alpha}}$, that are adjacent with $x$ and are connected to $g_2$ and $g_3$.
• Figure 2: On the left of the grey (shaded) area: Connection between some common branching vertices and a vertex $x$ in $X^i$ which corresponds to set $S = \{(\alpha, a'), (\beta, b'), (\gamma, c')\}$, for some $a', b', c' \in [n]$. The figure also shows vertices $x_{p^{\alpha}}$, $x_{q^{\alpha}}$, and $x_{r^{\alpha}}$. On the right of the grey (shaded) area: Connection between some common vertices and element-encoding gadget encoding $(\alpha, a)$ for some $a \in [n]$.

Theorems & Definitions (5)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5