Forcing Brushes
Dirk Meierling, Dieter Rautenbach

TL;DR
This paper provides concise proofs of inequalities relating the brushing number, zero forcing number, and line graph of a graph, simplifying previous complex proofs in graph theory.
Contribution
It introduces short and simple proofs for established inequalities involving brushing and zero forcing numbers in graphs and their line graphs.
Findings
Proved that B(G) ≤ Z(L(G)) for graphs without isolated vertices.
Proved that Z(G) ≤ Z(L(G)) for graphs without isolated vertices.
Simplified the understanding of relationships between graph invariants.
Abstract
We give short and simple proofs of the inequalities and first established by Erzurumluo\u{g}lu, Meagher, and Pike, where is a graph without isolated vertices, is the brushing number of , is the zero forcing number of , and is the line graph of .
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Complexity and Algorithms in Graphs
