Grundy dominating sequences and zero forcing sets

Brešar, Boštjan; Bujtás, Csilla; Gologranc, Tanja; Klavžar, Sandi; Košmrlj, Gašper; Patkós, Balázs; Tuza, Źsolt; Vizer, Máté

doi:10.1016/j.disopt.2017.07.001

Cited by 35 publications

(42 citation statements)

References 25 publications

(35 reference statements)

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“…(2) a dominating sequence, Proof. Case (1) is in [4]. Here we prove Case (4), as the others are similar.…”

Section: The Four Zero Forcing Type Parameters and The Four Grundy Dosupporting

confidence: 66%

Zero forcing number, Grundy domination number, and their variants

Lin

2019

Linear Algebra and its Applications

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This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem. We show that the Grundy domination type parameters are bounded above by the minimum rank type parameters. We also give a method to calculate the L-Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the L-Grundy domination number.

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“…(2) a dominating sequence, Proof. Case (1) is in [4]. Here we prove Case (4), as the others are similar.…”

Section: The Four Zero Forcing Type Parameters and The Four Grundy Dosupporting

confidence: 66%

Zero forcing number, Grundy domination number, and their variants

Lin

2019

Linear Algebra and its Applications

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“…While the length of a shortest legal dominating sequence is the domination number of G, the length of a longest one provides an upper bound for the size of dominating sets that can be constructed by a greedy domination procedure. Regarding algebraic properties, a strong connection between the Grundy domination number and the zero forcing number of a graph [1] was established in [6].…”

Section: Introductionmentioning

confidence: 99%

Grundy dominating sequences on X-join product

Nasini

Torres

2020

Discrete Applied Mathematics

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In this paper we study the Grundy domination number on the X-join product G ← R of a graph G and a family of graphs R = {Gv : v ∈ V (G)}. The results led us to extend the few known families of graphs where this parameter can be efficiently computed. We prove that if, for all v ∈ V (G), the Grundy domination number of Gv is given, and G is a power of a cycle, a power of a path, or a split graph, computing the Grundy domination number of G ← R can be done in polynomial time. In particular, the results for power of cycles and paths are derived from a polynomial reduction to the Maximum Weight Independent Set problem on these graphs.As a consequence, we derive closed formulas to compute the Grundy domination number of the lexicographic product G • H when G is a power of a cycle, a power of a path or a split graph, generalizing the results on cycles and paths given by Brešar et al. in 2016. Moreover, the results on the X-join product when G is a split graph also provide polynomial-time algorithms to compute the Grundy domination number for (q, q − 4) graphs, partner limited graphs and extended P4-laden graphs, graph classes which are high in the hierarchy of few P4's graphs.

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“…In the last decade, several Grundy domination invariants were introduced [7,11,12], which were motivated by the domination games introduced in [13,22]. An additional motivation for Grundy domination comes from the process of expanding a dominating set in a graph that is built on-line [10].…”

Section: Introductionmentioning

confidence: 99%

“…The maximum length of a Z-sequence in G is the Z-Grundy domination number, γ Z gr (G), of G. Note that if S is a Z-sequence, then saying that x ∈ S footprints a vertex y necessarily implies that vertices x and y are distinct. The Z-Grundy domination number was introduced in [7] as the dual of the zero forcing number. Notably, S is a Z-sequence if and only if the set of vertices outside S forms a zero forcing set [7].…”

Section: Introductionmentioning

confidence: 99%

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Grundy domination and zero forcing in Kneser graphs

2019

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Given a finite graph G, the maximum length of a sequence (v1,. .. , v k) of vertices in G such that each vi dominates a vertex that is not dominated by any vertex in {v1,. .. , vi−1} is called the Grundy domination number, γgr(G), of G. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that γgr(G) ≥ n+⌈ k 2 ⌉−2 k−1 holds for every connected k-regular graph of order n different from K k+1 and 2C4. The bound in the case k = 3 reduces to γgr(G) ≥ n 2 , and we characterize the connected cubic graphs with γgr(G) = n 2. If G is different from K4 and K3,3, then n 2 is also an upper bound for the zero forcing number of a connected cubic graph, and we characterize the connected cubic graphs attaining this bound.

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Grundy dominating sequences and zero forcing sets

Cited by 35 publications

References 25 publications

Zero forcing number, Grundy domination number, and their variants

Zero forcing number, Grundy domination number, and their variants

Grundy dominating sequences on X-join product

Grundy domination and zero forcing in Kneser graphs

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