Generalized limited packings of some graphs with a limited number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math>-partners

Dobson, Maria Patricia; Hinrichsen, Erica G.; Leoni, Valeria Alejandra

doi:10.1016/j.tcs.2014.11.014

Cited by 6 publications

(8 citation statements)

References 11 publications

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“…Here every vertex can have at most k neighbors in a k-limited packing set S. To achieve some bounds, a probabilistic approach to k-limited packings was introduced in [7]. A further generalization of it is shown in [3]. It brings a dynamic approach concerning the vertices of G, where dierent vertices can have a dierent number of neighbors in a generalized limited packing.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 99%

“…It brings a dynamic approach concerning the vertices of G, where dierent vertices can have a dierent number of neighbors in a generalized limited packing. As shown in [3], the problem of computing the packing number of graphs is NP-hard but polynomially solvable for P 4 -tidy graphs.…”

Section: Introductionmentioning

confidence: 99%

Incidence dimension and 2-packing number in graphs

et al. 2022

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“…A generalization of packing presented in [6] is the k-limited packing where every vertex can have at most k neighbors in a k-limited packing set S. A probabilistic approach to k-limited packings can be found in [5]. A further generalization, that is, the generalized limited packing of the k-limited packing, see [3], brings a dynamic approach with respect to the vertices of G, where v ∈ V (G) can have a different number of neighbors k v for every vertex v in a generalized limited packing. The problem of generalized limited packing is NP-complete, but solvable in polynomial time for P 4 -tidy graphs as shown in [3].…”

Section: Introductionmentioning

confidence: 99%

“…A further generalization, that is, the generalized limited packing of the k-limited packing, see [3], brings a dynamic approach with respect to the vertices of G, where v ∈ V (G) can have a different number of neighbors k v for every vertex v in a generalized limited packing. The problem of generalized limited packing is NP-complete, but solvable in polynomial time for P 4 -tidy graphs as shown in [3].…”

Section: Introductionmentioning

confidence: 99%

Graphs with unique maximum packing of closed neighborhoods

Božović¹,

Peterin²

2020

Discuss. Math. Graph Theory

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A packing of a graph G is a subset P of the vertex set of G such that the closed neighborhoods of any two distinct vertices of P do not intersect. We study graphs with a unique packing of the maximum cardinality. We present several general properties for such graphs. These properties are used to characterize the trees with a unique maximum packing. Two characterizations are presented where one of them is inductive based on five operations.

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“…More results on this topic can be found in [23]. A further generalization, that is, generalized limited packing of the k-limited packing (see [5]) brings a dynamic approach with respect to the vertices of G, where every v ∈ V (G) can have a different number of neighbors k v in a generalized limited packing. The problem is NP-complete, but solvable in polynomial time for P 4 -tidy graphs as shown in [5].…”

Section: Introductionmentioning

confidence: 99%

(Open) packing number of some graph products

Mojdeh,

Peterin,

Samadi

et al. 2019

Preprint

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The packing number of a graph G is the maximum number of closed neighborhoods of vertices in G with pairwise empty intersections. Similarly, the open packing number of G is the maximum number of open neighborhoods in G with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.

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Generalized limited packings of some graphs with a limited number of P4-partners

Cited by 6 publications

References 11 publications

Incidence dimension and 2-packing number in graphs

Incidence dimension and 2-packing number in graphs

Graphs with unique maximum packing of closed neighborhoods

(Open) packing number of some graph products

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