The mixed search game against an agile and visible fugitive is monotone

TL;DR

This paper studies a variant of the graph search game involving an agile, visible fugitive, proving the problem is in NP and introducing the concept of tight bramble as an obstruction.

Contribution

It establishes the monotonicity of the mixed search game against an agile and visible fugitive and characterizes the search number via the Cartesian tree product number.

Findings

The search game is monotone, simplifying strategy certification.

Deciding the search number is in NP, enabling efficient verification.

The search number equals the Cartesian tree product number of the graph.

Abstract

We consider the mixed search game against an agile and visible fugitive. This is the variant of the classic fugitive search game on graphs where searchers may be placed to (or removed from) the vertices or slide along edges. Moreover, the fugitive resides on the edges of the graph and can move at any time along unguarded paths. The mixed search number against an agile and visible fugitive of a graph $G$, denoted $avms(G)$, is the minimum number of searchers required to capture to fugitive in this graph searching variant. Our main result is that this graph searching variant is monotone in the sense that the number of searchers required for a successful search strategy does not increase if we restrict the search strategies to those that do not permit the fugitive to visit an already clean edge. This means that mixed search strategies against an agile and visible fugitive can be polynomially certified, and therefore that the problem of deciding, given a graph $G$ and an integer $k,$ whether $avms(G)\leq k$ is in NP. Our proof is based on the introduction of the notion of tight bramble, that serves as an obstruction for the corresponding search parameter. Our results imply that for a graph $G$, $avms(G)$ is equal to the Cartesian tree product number of $G$ that is the minimum $k$ for which $G$ is a minor of the Cartesian product of a tree and a clique on $k$ vertices.