Geodesic
Zero and total forcing dense graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 619-634

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If S is a set of colored vertices in a simple graph G, then one may allow a colored vertex with exactly one non-colored neighbor to force its non-colored neighbor to become colored. If by iteratively applying this color change rule, all of the vertices in G become colored, then S is a zero forcing set of G. The minimum cardinality of a zero forcing set in G, written Z(G), is the zero forcing number of G. If in addition, S induces a subgraph of G without isolated vertices, then S is a total forcing set of G. The total forcing number of G, written F_t(G), is the minimum cardinality of a total forcing set in G. In this paper we introduce, and study, the notion of graphs for which all vertices are contained in some minimum zero forcing set, or some minimum total forcing set; we call such graphs ZF-dense and TF-dense, respectively. A graph is ZTF-dense if it is both ZF-dense and TF-dense. We determine various classes of ZTF-dense graphs, including among others, cycles, complete multipartite graphs of order at least three that are not stars, wheels, n-dimensional hypercubes, and diamond-necklaces. We show that no tree of order at least three is ZTF-dense. We show that if G and H are connected graphs of order at least two that are both ZF-dense, then the join G + H of G and H is ZF-dense.
Keywords: zero forcing sets, zero forcing number, ZF-dense 
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Davila, Randy; Henning, Michael; Pepper, Ryan. Zero and total forcing dense graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 619-634. http://geodesic.mathdoc.fr/item/DMGT_2023_43_3_a2/