Reconfiguration graphs of zero forcing sets

Geneson, Jesse; Haas, Ruth; Hogben, Leslie

doi:10.48550/arxiv.2009.00220

Cited by 2 publications

(2 citation statements)

References 16 publications

(20 reference statements)

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“…In the seminal paper for zero forcing [2] it was proved that Z(T ) = P (T ) for any tree T , and consequently Z(F ) = P (F ) for any forest F ; see also [18]. Moreover, the set obtained from a path cover P of a tree T by taking one end-vertex of each path P ∈ P is a zero forcing set of T .…”

Section: Unique Zero Forcing Graphsmentioning

confidence: 99%

Graphs with unique zero forcing sets and Grundy dominating sets

Brešar,

Dravec

2021

Preprint

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The concept of zero forcing was introduced in the context of linear algebra, and was further studied by both graph theorists and linear algebraists. It is based on the process of activating vertices of a graph G starting from a set of vertices that are already active, and applying the rule that an active vertex with exactly one non-active neighbor forces that neighbor to become active. A set S ⊂ V (G) is called a zero forcing set of G if initially only vertices of S are active and the described process enforces all vertices of G to become active. The size of a minimum zero forcing set in G is called the zero forcing number of G. While a minimum zero forcing set can only be unique in edgeless graphs, we consider the weaker uniqueness condition, notably that for every two minimum zero forcing sets in a graph G there is an automorphism that maps one to the other. We characterize the class of trees that enjoy this condition by using properties of minimum path covers of trees. In addition, we investigate both variations of uniqueness for several concepts of Grundy domination, which first appeared in the context of domination games, yet they are also closely related to zero forcing. For each of the four variations of Grundy domination we characterize the graphs that have only one Grundy dominating set of the given type, and characterize those forests that enjoy the weaker (isomorphism based) condition of uniqueness. The latter characterizations lead to efficient algorithms for recognizing the corresponding classes of forests.

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Section: Unique Zero Forcing Graphsmentioning

confidence: 99%

Graphs with unique zero forcing sets and Grundy dominating sets

Brešar,

Dravec

2021

Preprint

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“…Maximum minimal sets and minimum maximal sets have been studied in the context of many other graph parameters, including independent sets (see [17,19]), dominating sets (see [5,6]), matchings (see [15,16]), and vertex covers (see [8,25]). Studying minimal zero forcing sets can lead to a better understanding of the zero forcing process, e.g., in the context of zero forcing polynomials [9] and zero forcing reconfiguration graphs [18].…”

Section: Introductionmentioning

confidence: 99%

Minimal Zero Forcing Sets

Brimkov¹,

Carlson²

2022

Preprint

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In this paper, we study minimal (with respect to inclusion) zero forcing sets. We first investigate when a graph can have polynomially or exponentially many distinct minimal zero forcing sets. We also study the maximum size of a minimal zero forcing set Z(G), and relate it to the zero forcing number Z(G). Surprisingly, we show that the equality Z(G) = Z(G) is preserved by deleting a universal vertex, but not by adding a universal vertex. We also characterize graphs with extreme values of Z(G) and explore the gap between Z(G) and Z(G).

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Reconfiguration graphs of zero forcing sets

Cited by 2 publications

References 16 publications

Graphs with unique zero forcing sets and Grundy dominating sets

Graphs with unique zero forcing sets and Grundy dominating sets

Minimal Zero Forcing Sets

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