Zero forcing in iterated line digraphs

TL;DR

This paper investigates zero forcing in iterated line digraphs, establishing relationships with power domination, and determines optimal parameters for regular cases, with applications to common digraph families.

Contribution

It introduces new relationships between zero forcing and power domination in iterated line digraphs and provides exact parameters for regular cases.

Findings

Regular iterated line digraphs have optimal zero forcing and power domination parameters.

Explicit constructions for minimum rank and nullity are provided.

Results apply to digraph families used in practical applications.

Abstract

Zero forcing is a propagation process on a graph, or digraph, defined in linear algebra to provide a bound for the minimum rank problem. Independently, zero forcing was introduced in physics, computer science and network science, areas where line digraphs are frequently used as models. Zero forcing is also related to power domination, a propagation process that models the monitoring of electrical power networks. In this paper we study zero forcing in iterated line digraphs and provide a relationship between zero forcing and power domination in line digraphs. In particular, for regular iterated line digraphs we determine the minimum rank/maximum nullity, zero forcing number and power domination number, and provide constructions to attain them. We conclude that regular iterated line digraphs present optimal minimum rank/maximum nullity, zero forcing number and power domination number, and apply our results to determine those parameters on some families of digraphs often used in applications.