On the Complexity of the Positive Semidefinite Zero Forcing Number

TL;DR

This paper explores the computational complexity of the positive semidefinite zero forcing number, establishing NP-completeness results and providing a linear time algorithm for chordal graphs.

Contribution

It introduces a relation between zero forcing and fast-mixed searching, and presents a linear time algorithm for chordal graphs' positive zero forcing number.

Findings

NP-completeness of zero forcing problem

Linear time algorithm for chordal graphs

NP-completeness of zero forcing set with additional properties

Abstract

The positive zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring, and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and the fast-mixed searching, which implies some NP-completeness results for the zero forcing problem. For chordal graphs much is understood regarding the relationships between positive zero forcing and clique coverings. Building upon constructions associated with optimal tree covers and forest covers, we present a linear time algorithm for computing the positive zero forcing number of chordal graphs. We also prove that it is NP-complete to determine if a graph has a positive zero forcing set with an additional property.