Graphs with unique maximum packing of closed neighborhoods

Božović, Dragana; Peterin, Iztok

doi:10.7151/dmgt.2304

Cited by 4 publications

(4 citation statements)

References 13 publications

(15 reference statements)

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“…In another paper of Gunther et al [19] the trees having a unique minimum dominating set were characterized, while Haynes and Henning in [21] characterized the trees with a unique minimum total dominating set. Two recent papers considered graphs, and in particular trees, that have unique maximum (open) packings [5,16]. In addition, it was proved in [16] that the recognition of the graphs with a unique maximum (open) packing is polynomially equivalent to the recognition of the graphs with a unique maximum independent set, and that the complexity of all three problems is not polynomial, unless P=NP [16].…”

Section: Introductionmentioning

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: Introductionmentioning

confidence: 99%

Graphs with unique zero forcing sets and Grundy dominating sets

Brešar,

Dravec

2021

Preprint

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“…Recently, Jaume and Molina [11] provided a characterization of such trees from an algebraic point of view, which can be used for efficient recognition of trees with a unique maximum independent set; see Section 3. Trees that have a unique maximum 2-packing (where 2-packing is a set of vertices having pairwise disjoint closed neighborhoods) were also characterized recently [1]. In this paper, we begin the study of graphs which have a unique maximum open packing, and we denote the set of such graphs by C ρ o .…”

Section: Introductionmentioning

confidence: 99%

Graphs with a unique maximum open packing

Brešar,

Kuenzel,

Rall

2019

Preprint

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A set S of vertices in a graph is an open packing if (open) neighborhoods of any two distinct vertices in S are disjoint. In this paper, we consider the graphs that have a unique maximum open packing. We characterize the trees with this property by using four local operations such that any nontrivial tree with a unique maximum open packing can be obtained by a sequence of these operations starting from P 2 . We also prove that the decision version of the open packing number is NP-complete even when restricted to graphs of girth at least 6. Finally, we show that the recognition of the graphs with a unique maximum open packing is polynomially equivalent to the recognition of the graphs with a unique maximum independent set, and we prove that the complexity of both problems is not polynomial, unless P=NP.

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“…We next provide an exact result for the class of graphs with the unique maximum packing. Two characterizations of trees with the unique maximum packing were presented recently in [2]. Theorem 8.…”

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confidence: 99%

Incidence dimension and 2-packing number in graphs

et al. 2022

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Abstract. Let G = (V,E) be a graph. A set of vertices A is an incidence generator for G if for any two distinct edges e,f ∈ E(G) there exists a vertex from A which is an endpoint of either e or f. The smallest cardinality of an incidence generator for G is called the incidence dimension and is denoted by dimI(G). A set of vertices P ⊆ V (G) is a 2-packing of G if the distance in G between any pair of distinct vertices from P is larger than two. The largest cardinality of a 2-packing of G is the packing number of G and is denoted by ρ(G). In this article, the incidence dimension is introduced and studied. The given results show a close relationship between dimI(G) and ρ(G). We rst note that the complement of any 2-packing in graph G is an incidence generator for G, and further show that either dimI(G) = ρ(G) or dimI(G) = ρ(G)−1 for any graph G. In addition, we present some bounds for dimI(G) and prove that the problem of determining the incidence dimension of a graph is NP-hard.

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Graphs with unique maximum packing of closed neighborhoods

Cited by 4 publications

References 13 publications

Graphs with unique zero forcing sets and Grundy dominating sets

Graphs with unique zero forcing sets and Grundy dominating sets

Graphs with a unique maximum open packing

Incidence dimension and 2-packing number in graphs

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