Contraction Obstructions for Connected Graph Searching

TL;DR

This paper characterizes the finite obstruction set of graphs with connected mixed search number at most 2, revealing the importance of search direction and providing a comprehensive understanding of these graph classes.

Contribution

The paper explicitly determines the finite obstruction set for graphs with connected mixed search number at most 2, consisting of 177 graphs, and discusses the complexity of obstruction sets for higher values.

Findings

The obstruction set for k=2 is finite and contains 177 graphs.

The sense of direction in search strategies affects connected search outcomes.

A double exponential lower bound exists for the size of obstruction sets when finite.

Abstract

We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for $k=2$, where $k$ is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite.