Zero forcing versus domination in cubic graphs

Randy Davila () and Michael A. Henning ()
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Randy Davila: University of Johannesburg
Michael A. Henning: University of Johannesburg

Journal of Combinatorial Optimization, 2021, vol. 41, issue 2, No 16, 553-577

Abstract: Abstract In this paper, we study a dynamic coloring of the vertices of a graph G that 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 a zero forcing set of G if, by iteratively5 applying the forcing process, every vertex in G becomes colored. The zero forcing number of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if $$G \ne K_4$$ G ≠ K 4 is a connected cubic graph, then the zero forcing number of G is bounded above by twice its domination number, where the domination number of G is the minimum cardinality of a set of vertices of G such that every vertex not in S is adjacent to some vertex in S.

Keywords: Zero forcing set; Zero forcing number; Dominating set; Domination number; 05C69 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10878-020-00692-z

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