Complexity results for a cops and robber game on directed graphs

TL;DR

It is proved that deciding whether the cop number of a digraph is equal to 1 is NP‐hard, whereas this is decidable in polynomial time for tournaments and it is shown that computing the cop number for general digraphs is fixed parameter tractable when parameterized by a generalization of vertex cover.

Abstract

We investigate a cops and robber game on directed graphs, where the robber moves along the arcs of the graph, while the cops can select any position at each time step. Our main focus is on the cop number: the minimum number of cops required to guarantee the capture of the robber. We prove that deciding whether the cop number of a digraph is equal to 1 is NP-hard, whereas this is decidable in polynomial time for tournaments. Furthermore, we show that computing the cop number for general digraphs is fixed parameter tractable when parameterized by a generalization of vertex cover. However, for tournaments, tractability is achieved with respect to the minimum size of a feedback vertex set. Among our findings, we prove that the cop number of a digraph is equal to that of its reverse digraph, and we draw connections to the matrix mortality problem.

Figures (4)

• Figure 1: Reduction from $3$-partition: $a_1 = a_2 =4$, $a_3 = a_4 = a_5 = a_6 = 3$, $\beta = 10$, $n = 6$,$m = 2$, $|V_i|=a_i, 1 \leq i \leq 6$, $|I_j| = \beta, 1 \leq j \leq 3$. Thick arrows indicate all arcs from a set to the other
• Figure 2: Illustration of subdivision: $k = 2$, $D$ on the left and $\tilde{D}$ on the right
• Figure 3: The digraph related to the proof of Theorem \ref{['th:cn=1']}
• Figure 4: A $2$-copwin tournament of order $3n-2$ (here $n=4$) whose minimum feedback vertex set is of size $n$ (a thick arrow from $C_i$ to $C_j$ represents all arcs from $C_i$ vertices to $C_j$ vertices)

Theorems & Definitions (45)

• proof
• Corollary 3.2
• Corollary 3.3
• Lemma 3.4
• proof
• Theorem 4.1
• proof
• Claim 1
• proof
• Theorem 4.2