Compressed Cliques Graphs, Clique Coverings and Positive Zero Forcing

TL;DR

This paper explores positive zero forcing in graphs through clique coverings, focusing on the properties of compressed cliques graphs and their connections to other graph structures.

Contribution

It introduces and analyzes the properties of compressed cliques graphs in relation to positive zero forcing and clique coverings, expanding understanding of these graph parameters.

Findings

Compressed cliques graphs have unique properties.

Certain forbidden subgraphs are identified.

Connections to Johnson graphs are established.

Abstract

Zero forcing parameters, associated with graphs, have been studied for over a decade, and have gained popularity as the number of related applications grows. In particular, it is well-known that such parameters are related to certain vertex coverings. Continuing along these lines, we investigate positive zero forcing within the context of certain clique coverings. A key object considered here is the compressed cliques graph. We study a number of properties associated with the compressed cliques graph, including: uniqueness, forbidden subgraphs, connections to Johnson graphs, and positive zero forcing.