Zero blocking numbers of graphs with complexity results

TL;DR

This paper investigates the zero blocking number in graphs, providing exact values for unions, joins, hypercubes, and an efficient algorithm for trees, advancing understanding of blocking sets in graph theory.

Contribution

It introduces new results for zero blocking numbers in graph unions, joins, hypercubes, and offers a linear-time algorithm for trees.

Findings

Zero blocking numbers for unions and joins of graphs are characterized.

All minimum zero blocking sets in hypercubes are identified.

A linear-time algorithm for computing zero blocking numbers in trees is developed.

Abstract

For a graph $G$ in which vertices are either black or white, a zero forcing process is an iterative vertex color changing process such that the only white neighbor of a black vertex becomes black in the next time step. A zero forcing set is an initial subset of black vertices in a zero forcing process ultimately expands to include all vertices of the graph; otherwise we call its complement a zero blocking set. The zero blocking number $B(G)$ of $G$ is the minimum size of a zero blocking set. This paper determines zero blocking numbers of the union and the join of two graphs. It also determines all minimum zero blocking sets of hypercubes. Finally, a linear-time algorithm for the zero blocking numbers of trees is given.