- Tytuł:
- Zero and total forcing dense graphs
- Autorzy:
-
Davila, Randy
Henning, Michael
Pepper, Ryan - Tematy:
-
zero forcing sets
zero forcing number
ZF-dense - Pokaż więcej
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Powiązania:
- https://bibliotekanauki.pl/articles/59898585.pdf Link otwiera się w nowym oknie
- Opis:
- 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.
- Dostawca treści:
- Biblioteka Nauki
Artykuł