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Zero forcing versus domination in cubic graphs

Author

Listed:
  • Randy Davila

    (University of Johannesburg
    University of Houston–Downtown)

  • Michael A. Henning

    (University of Johannesburg)

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.

Suggested Citation

  • Randy Davila & Michael A. Henning, 2021. "Zero forcing versus domination in cubic graphs," Journal of Combinatorial Optimization, Springer, vol. 41(2), pages 553-577, February.
    • Handle: RePEc:spr:jcomop:v:41:y:2021:i:2:d:10.1007_s10878-020-00692-z
    DOI: 10.1007/s10878-020-00692-z
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    References listed on IDEAS

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    1. Daniela Ferrero & Cyriac Grigorious & Thomas Kalinowski & Joe Ryan & Sudeep Stephen, 2019. "Minimum rank and zero forcing number for butterfly networks," Journal of Combinatorial Optimization, Springer, vol. 37(3), pages 970-988, April.
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