The multiple domination and limited packing problems in graphs

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

doi:10.1016/j.ipl.2011.09.002

Cited by 24 publications

(17 citation statements)

References 15 publications

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“…(). 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%

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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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“…(). 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 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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“…Let (Liao and Chang, 2003) P (Lee and Chang, 2008) Doubly chordal NP-c (Lee and Chang, 2008) P (Lee and Chang, 2008) Dually chordal NP-c (Lee and Chang, 2008) P (Lee and Chang, 2008) Cographs P (Lee and Chang, 2008) P (Lee and Chang, 2008) P 4 -tidy P (Dobson et al, 2011) P (Argiroffo et al, 2015) Bounded cliquewidth P (Argiroffo et al, 2015) P (Argiroffo et al, 2015 k-tuple dominating function of G is called the k-tuple domination number of G and is denoted by γ ×k (G) (Harary and Haynes, 1996).…”

Section: Introductionmentioning

confidence: 99%

On the complexity of the labeled domination problem in graphs

Argiroffo

Leoni

Torres

2016

Int Trans Operational Res

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In 2008, a unified approach (labeled domination) to several domination problems (k‐tuple domination, {k}‐domination, and M‐domination, among others) was introduced. The labeled domination problem is to find an L‐dominating function of minimum weight in a graph. It is an NP‐complete problem even when restricted to split graphs and bipartite graphs. On the other hand, it is known to be polynomial‐time solvable for the class of strongly chordal graphs. In this paper, we state explicit formulas that relate the domination numbers considered. These relationships allow us to enlarge the family of graphs where the labeled domination problem is polynomial‐time solvable to the class of graphs having cliquewidth bounded by a constant.

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“…The problem of finding a 2-packing (1-limited packing) of maximum size is shown to be NP -complete by Hochbaum and Schmoys [8]. In [4], it is shown that the problem of finding a maximum size k-limited packing is NP -complete even for the classes of split and bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

“…2-Packings (1-limited packings) are well-studied in the literature from the structural and algorithmic point of view (e.g., see [8,9,10,11]) and in connection with other graph parameters (e.g., see [3,7,9,10,12]). In particular, several papers discuss connections between packings and dominating sets in graphs (e.g., see [3,4,6,7,10]). Although the formal definitions for packings and dominating sets may appear to be similar, the problems have a very different nature: one of the problems is a maximization problem not to break some (security) constraints, and the other is a minimization problem to satisfy some reliability requirements.…”

Section: Introductionmentioning

confidence: 99%

The probabilistic approach to limited packings in graphs

Gagarin

Zverovich

2015

Discrete Applied Mathematics

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We consider (closed neighbourhood) packings and their generalization in graphs.of a graph G is the largest size of a k-limited packing in G. Limited packing problems can be considered as secure facility location problems in networks.In this paper, we develop a new probabilistic approach to limited packings in graphs, resulting in lower bounds for the k-limited packing number and a randomized algorithm to find k-limited packings satisfying the bounds. In particular, we prove that for any graph G of order n with maximum vertex degree ∆,.The problem of finding a maximum size k-limited packing is known to be N Pcomplete even in split or bipartite graphs.

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The multiple domination and limited packing problems in graphs

Cited by 24 publications

References 15 publications

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

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

On the complexity of the labeled domination problem in graphs

The probabilistic approach to limited packings in graphs

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