Zero forcing number, constrained matchings and strong structural controllability

TL;DR

This paper proves the NP-hardness of computing the zero forcing number for directed graphs with loops and establishes a zero forcing set criterion for strong controllability in networked systems, including efficient solutions for certain cases.

Contribution

It extends NP-hardness results to directed graphs with loops and links zero forcing sets to strong controllability, offering new theoretical insights and practical algorithms.

Findings

NP-hardness of zero forcing number for directed graphs with loops

Zero forcing sets characterize strong controllability in directed graphs

Efficient method for minimal input set in tree-structured systems

Abstract

The zero forcing number is a graph invariant introduced to study the minimum rank of the graph. In 2008, Aazami proved the NP-hardness of computing the zero forcing number of a simple undirected graph. We complete this NP-hardness result by showing that the non-equivalent problem of computing the zero forcing number of a directed graph allowing loops is also NP-hard. The rest of the paper is devoted to the strong controllability of a networked system. This kind of controllability takes into account only the structure of the interconnection graph, but not the interconnection strengths along the edges. We provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops. Moreover, we explain how our result differs from a recent related result discovered by Monshizadeh et al. Finally, we show how to solve the problem of finding efficiently a minimum-size input set for the strong controllability of a self-damped system with a tree-structure.