Algorithmic aspects of open neighborhood location–domination in graphs

Panda, B. S.; Pandey, Arti

doi:10.1016/j.dam.2015.03.002

Cited by 7 publications

(8 citation statements)

References 9 publications

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“…The DECIDE MIN OLD-SET problem is known to be NPcomplete for bipartite graphs [12]. In this section, we first improve this complexity result by proving that the DECIDE OLD-SET problem is NP-complete for perfect elimination bipartite graphs, a subclass of bipartite graphs.…”

Section: Old-set In Bipartite Graphsmentioning

confidence: 99%

“…The concept of open location-domination number was introduced by Seo and Slater in [14] and further studied in [2], [12], [15], [16]. The DECIDE MIN OLD-SET problem is NPcomplete for graphs [14], bipartite graphs [12], doubly chordal graphs [12], and split graphs [12].…”

mentioning

confidence: 99%

“…The DECIDE MIN OLD-SET problem is NPcomplete for graphs [14], bipartite graphs [12], doubly chordal graphs [12], and split graphs [12]. This problem is also NPcomplete for the graphs which are at the same time interval graphs and permutation graphs [3].…”

mentioning

confidence: 99%

“…This problem is also NPcomplete for the graphs which are at the same time interval graphs and permutation graphs [3]. On the positive side, the MIN OLD-SET problem can be solved in linear time for paths [14], trees [12], and cographs [3]. In this paper, we further study the algorithmic aspects of the MIN OLD-SET problem.…”

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confidence: 99%

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Open Neighborhood Locating-Dominating Set in Graphs: Complexity and Algorithms

Pandey

2015

2015 International Conference on Information Technology (ICIT)

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A set D ⊆ V of a graph G = (V, E) is called an open neighborhood locating-dominating set (OLD-set) ifGiven a graph G = (V, E), the MIN OLD-SET problem is to find an OLD-set of minimum cardinality. The cardinality of a minimum OLD-set of G is called the open neighborhood location-domination number of G, and is denoted by γ old (G). Given a graph G and a positive integer k, the DECIDE OLD-SET problem is to decide whether G has an OLDset of cardinality at most k. The DECIDE OLD-SET problem is known to be NP-complete for bipartite graphs. In this paper, we strengthen this NP-complete result by showing that the DECIDE OLD-SET problem remains NP-complete for perfect elimination bipartite graphs, a subclass of bipartite graphs. Then, we show that the MIN OLD-SET problem can be solved in polynomial time in chain graphs, a subclass of perfect elimination bipartite graphs. We show that for a graph G, γ old (G) ≥ 2n Δ(G)+2 , where n denotes the number of vertices in G, and Δ(G) denotes the maximum degree of G. As a consequence we obtain a Δ(G)+2 2 -approximation algorithm for the MIN OLD-SET problem. Finally, we prove that the MIN OLD-SET problem is APX-complete for chordal graphs with maximum degree 4.

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Section: Old-set In Bipartite Graphsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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Open Neighborhood Locating-Dominating Set in Graphs: Complexity and Algorithms

Pandey

2015

2015 International Conference on Information Technology (ICIT)

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“…The open locating-dominating number γ OLD (G) of a graph G is the minimum cardinality of an OLD-code of G. Determining γ OLD (G) is in general NP-hard [22] and remains NP-hard for perfect elimination bipartite graphs and APXcomplete for chordal graphs with maximum degree 4 [18]. Closed formulas for the exact value of γ OLD (G) have been found so far only for restricted graph families such as cliques and paths [22], some algorithmic aspects are discussed in [19]. More results on all three problems are listed in [16].…”

Section: Introductionmentioning

confidence: 99%

Linear-time algorithms for three domination-based separation problems in block graphs

Argiroffo

Bianchi

Lucarini

et al. 2020

Discrete Applied Mathematics

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The problems of determining minimum identifying, locating-dominating or open locating-dominating codes are special search problems that are challenging both from a theoretical and a computational point of view, even for several graph classes where other in general hard problems are easy to solve, like bipartite graphs or chordal graphs. Hence, a typical line of attack for these problems is to determine minimum codes of special graphs. In this work we study the problem of determining the cardinality of minimum such codes in block graphs (that are diamond-free chordal graphs). We present linear-time algorithms for these problems, as a generalization of a linear-time algorithm proposed by Auger in 2010 for identifying codes in trees. Thereby, we provide a subclass of chordal graphs for which all three problems can be solved in linear time.

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