Algorithms and Hardness for Geodetic Set on Tree-like Digraphs

TL;DR

This paper studies the computational complexity of the Geodetic Set problem on directed graphs, providing polynomial algorithms for certain tree-like structures and proving NP-hardness in more general cases.

Contribution

It introduces polynomial-time algorithms for Geodetic Set on ditrees and fixed-parameter algorithms based on feedback edge set number, while establishing NP-hardness on DAGs with specific constraints.

Findings

Polynomial-time algorithm for Geodetic Set on ditrees.

Fixed-parameter tractability when underlying undirected graph has bounded feedback edge set number.

NP-hardness of Geodetic Set on DAGs with constant feedback vertex set number and pathwidth.

Abstract

In the GEODETIC SET problem, an input is a (di)graph $G$ and integer $k$, and the objective is to decide whether there exists a vertex subset $S$ of size $k$ such that any vertex in $V(G)\setminus S$ lies on a shortest (directed) path between two vertices in $S$. The problem has been studied on undirected and directed graphs from both algorithmic and graph-theoretical perspectives. We focus on directed graphs and prove that GEODETIC SET admits a polynomial-time algorithm on ditrees, that is, digraphs with possible 2-cycles when the underlying undirected graph is a tree (after deleting possible parallel edges). This positive result naturally leads us to investigate cases where the underlying undirected graph is "close to a tree". Towards this, we show that GEODETIC SET on digraphs without 2-cycles and whose underlying undirected graph has feedback edge set number $\textsf{fen}$, can be solved in time $2^{\mathcal{O}(\textsf{fen})} \cdot n^{\mathcal{O}(1)}$, where $n$ is the number of vertices. To complement this, we prove that the problem remains NP-hard on DAGs (which do not contain 2-cycles) even when the underlying undirected graph has constant feedback vertex set number and constant pathwidth. Our last result significantly strengthens the result of Ara\'ujo and Arraes [Discrete Applied Mathematics, 2022] that the problem is NP-hard on DAGs when the underlying undirected graph is either bipartite, cobipartite or split.