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Total Forcing and Zero Forcing in Claw-Free Cubic Graphs

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Abstract

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set (zero forcing set) of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The total forcing number of G, denoted \(F_t(G)\), is the minimum cardinality of a total forcing set in G. We prove that if G is a connected, claw-free, cubic graph of order \(n \ge 6\), then \(F_t(G) \le \frac{1}{2}n\), where a claw-free graph is a graph that does not contain \(K_{1,3}\) as an induced subgraph. The graphs achieving equality in these bounds are characterized.

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Acknowledgements

The authors express their sincere thanks to the anonymous referees for their helpful comments and suggestions that improved the exposition and presentation of the manuscript.

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Correspondence to Michael A. Henning.

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Davila, R., Henning, M.A. Total Forcing and Zero Forcing in Claw-Free Cubic Graphs. Graphs and Combinatorics 34, 1371–1384 (2018). https://doi.org/10.1007/s00373-018-1934-4

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  • DOI: https://doi.org/10.1007/s00373-018-1934-4

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