The k-limited packing and k-tuple domination problems in strongly chordal, P4-tidy and split graphs

Dobson, Maria Patricia; Leoni, Valeria Alejandra; Nasini, Graciela L.

doi:10.1016/j.endm.2010.05.071

Cited by 13 publications

(4 citation statements)

References 9 publications

(20 reference statements)

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“…Nevertheless, for the following graph these numbers do not coincide: The above definitions induce the study -started in [3] and [5]-of the computational complexity of the following decision problems:…”

Section: Basic Definitions and Preliminariesmentioning

confidence: 99%

Towards a Polynomial Equivalence Between $$\{k\}$$ -Packing Functions and k-Limited Packings in Graphs

Leoni

Dobson

2016

Lecture Notes in Computer Science

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Given a positive integer k, the {k}-packing function problem ({k}PF) is to find in a given graph G, a function f of maximum weight that assigns a non-negative integer to the vertices of G in such a way that the sum of f (v) over each closed neighborhood is at most k. This notion was recently introduced as a variation of the k-limited packing problem (kLP) introduced in 2010, where the function was supposed to assign a value in {0, 1}. For all the graph classes explored up to now, {k}PF and kLP have the same computational complexity. It is an open problem to determine a graph class where one of them is NP-complete and the other, polynomially solvable. In this work, we first prove that {k}PF is NP-complete for bipartite graphs, as kLP is known to be. We also obtain new graph classes where the complexity of these problems would coincide.

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Section: Basic Definitions and Preliminariesmentioning

confidence: 99%

Towards a Polynomial Equivalence Between $$\{k\}$$ -Packing Functions and k-Limited Packings in Graphs

Leoni

Dobson

2016

Lecture Notes in Computer Science

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“…Gallant et al in [8] introduced k-limited packing as a modification of packing number problem allowing that intersection of each closed neighborhood with a given set contains no more than k vertices. In [5,6] Dobson et al proved that k-limited packing is NP-complete for split and bipartite graphs. It was also shown that P 4 -tidy graphs are solvable in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Some properties of $\{k\}$-packing function problem in graphs

Kratica¹,

Savić²,

Maksimović³

2018

Preprint

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The recently introduced {k}-packing function problem is considered in this paper. Special relation between a case when k = 1, k ≥ 2 and linear programming relaxation is introduced with sufficient conditions for optimality. For arbitrary simple connected graph G there is construction procedure for finding values of k for which L {k} (G) can be determined in the polynomial time. Additionally, relationship between {1}-packing function and independent set number is established. Optimal values for some special classes of graphs and general upper and lower bounds are introduced.

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“…The study of the computational complexity of this problem was started in Dobson et al. (). Regarding NP‐completeness results, the graph is considered NP‐complete even for instances given by split graphs (Dobson et al., ), and also for bipartite graphs (Dobson et al., ).…”

Section: Introductionmentioning

confidence: 99%

“…The above definition induces the study of the problem of deciding if a given graph G has a k-limited packing of weight L k (G). The study of the computational complexity of this problem was started in Dobson et al (2010). Regarding NP-completeness results, the graph is considered NP-complete even for instances given by split graphs (Dobson et al, 2010), and also for bipartite graphs (Dobson et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

On the complexity of the {k}‐packing function problem

Dobson

Hinrichsen

Leoni

2016

Int Trans Operational Res

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Given a positive integer k, the "{k}-packing function problem" ({k}PF) is to find in a given graph G, a function f that assigns a nonnegative integer to the vertices of G in such a way that the sum of f (v) over each closed neighborhood is at most k and over the whole vertex set of G (weight of f ) is maximum. It is known that {k}PF is linear time solvable in strongly chordal graphs and in graphs with clique-width bounded by a constant. In this paper we prove that {k}PF is NP-complete, even when restricted to chordal graphs that constitute a superclass of strongly chordal graphs. To find other subclasses of chordal graphs where {k}PF is tractable, we prove that it is linear time solvable for doubly chordal graphs, by proving that it is so in the superclass of dually chordal graphs, which are graphs that have a maximum neighborhood ordering.

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The k-limited packing and k-tuple domination problems in strongly chordal, P4-tidy and split graphs

Cited by 13 publications

References 9 publications

Towards a Polynomial Equivalence Between $$\{k\}$$ -Packing Functions and k-Limited Packings in Graphs

Towards a Polynomial Equivalence Between $$\{k\}$$ -Packing Functions and k-Limited Packings in Graphs

Some properties of $\{k\}$-packing function problem in graphs

On the complexity of the {k}‐packing function problem

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