Deduction, Constrained Zero Forcing, and Constrained Searching

TL;DR

This paper introduces a new deduction-based graph searching process and demonstrates its equivalence in searcher requirements to other known processes, providing structural insights, complexity results, and properties of the process.

Contribution

It establishes the equivalence of deduction with zero forcing and fast-mixed search in terms of searcher number, and offers new bounds, structural characterizations, and complexity analyses.

Findings

Deduction requires the same number of searchers as zero forcing and fast-mixed search.

The paper provides structural characterizations and bounds for the deduction process.

NP-completeness is shown for computing the deduction parameter on arbitrary graphs.

Abstract

Deduction is a recently introduced graph searching process in which searchers clear the vertex set of a graph with one move each, with each searcher's movement determined by which of its neighbors are protected by other searchers. In this paper, we show that the minimum number of searchers required to clear the graph is the same in deduction as in constrained versions of other previously studied graph processes, namely zero forcing and fast-mixed search. We give a structural characterization, new bounds and a spectrum result on the number of searchers required. We consider the complexity of computing this parameter, giving an NP-completeness result for arbitrary graphs, and exhibiting families of graphs for which the parameter can be computed in polynomial time. We also describe properties of the deduction process related to the timing of searcher movement and the success of terminal layouts.