Algorithms and Hardness for Geodetic Set on Tree-like Digraphs

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. Our last result significantly strengthens the result of Araújo 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.

Figures (4)

• Figure 1: An example of a ditree $T$ and its contracted ditree $T^c$.
• Figure 2: In this example digraph $D$, the red arcs form a hanging ditree. The circled blue subgraph is the core digraph of $D$. It is composed of three core vertices in black, and five core oriented paths. In particular, the path $uvw$ is a core dipath.
• Figure 3: A partial representation of the digraph $D$ constructed during the reduction. Dashed arcs represent paths of length greater than one. Red arcs between a vertex and a vertex set mean that there exists such an arc for all vertices in the vertex set. Filled vertices are extremal vertices of $D$, that belong to any geodetic set. Arcs adjacent to vertices $\delta_i$ are represented for only one edge vertex and vertices associated with three different elements, each of them belonging to a different partition set of $U$. Pending vertices to $\delta_i$ and paths from edge vertices to $\delta_i$ are not represented.
• Figure 4: Details of the gadget ensuring adjacency of edges and elements. One edge vertex $u^e$ is represented, as well as three gadgets that each correspond to an element of a set $X^\delta$. Dashed arcs represent paths, whose length is indicated on the drawing. Filled vertices are extremal vertices that belong to any geodetic set of the digraph.

Theorems & Definitions (36)

• Proposition 1: AraujoA22, Proposition 6.1
• theorem 1
• theorem 2
• theorem 3
• Lemma 1: AraujoA22
• theorem 3
• Definition 1
• Lemma 2
• proof
• Lemma 3