An O(n)-Time Algorithm for the Paired-Domination Problem on Permutation Graphs

Lappas, E.; Nikolopoulos, Stavros D.; Palios, Leonidas

doi:10.1007/978-3-642-10217-2_36

Cited by 13 publications

(8 citation statements)

References 28 publications

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“…It is proven that, the decision version of the problem is NP-complete when restricted to special graph classes, including bipartite graphs [4], perfect elimination bipartite graphs [17], and split graphs [4]. But, on the good side, the problem is efficiently solvable in several important graph classes, including permutation graphs [13], interval graphs [4], block graphs [4], strongly chordal graphs [5], circular-arc graphs [14] and some others. A detailed survey of the results on paired domination can be found in [6].…”

Section: Decide Pd-set Problemmentioning

confidence: 99%

Complexity of Paired Domination in AT-free and Planar Graphs

Tripathi¹,

Kloks²,

Pandey³

et al. 2021

Preprint

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For a graph G = (V, E), a subset D of vertex set V , is a dominating set of G if every vertex not in D is adjacent to atleast one vertex of D. A dominating set D of a graph G with no isolated vertices is called a paired dominating set (PD-set), if G[D], the subgraph induced by D in G has a perfect matching. The MIN-PD problem requires to compute a PD-set of minimum cardinality. The decision version of the MIN-PD problem remains NP-complete even when G belongs to restricted graph classes such as bipartite graphs, chordal graphs etc. On the positive side, the problem is efficiently solvable for many graph classes including intervals graphs, strongly chordal graphs, permutation graphs etc. In this paper, we study the complexity of the problem in AT-free graphs and planar graph. The class of AT-free graphs contains cocomparability graphs, permutation graphs, trapezoid graphs, and interval graphs as subclasses. We propose a polynomial-time algorithm to compute a minimum PD-set in AT-free graphs. In addition, we also present a linear-time 2-approximation algorithm for the problem in AT-free graphs. Further, we prove that the decision version of the problem is NPcomplete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput.

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Section: Decide Pd-set Problemmentioning

confidence: 99%

Complexity of Paired Domination in AT-free and Planar Graphs

Tripathi¹,

Kloks²,

Pandey³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The class of permutation graphs behaves as the class of interval graphs in the sense that, on one hand, permutation graphs can be recognized in linear time [32] and, on the other hand, many NP-complete problems can be solved efficiently when the input is restricted to permutation graphs [8,33]. A graph G is a circle graph if G is the intersection model of a collection of chords in a circle (see Figure 2).…”

Section: Geometric Intersection Graph Classesmentioning

confidence: 99%

Distributed Interactive Proofs for the Recognition of Some Geometric Intersection Graph Classes

Jáuregui¹,

Montealegre²,

Rapaport³

2021

Preprint

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A graph G = (V, E) is a geometric intersection graph if every node v ∈ V is identified with a geometric object of some particular type, and two nodes are adjacent if the corresponding objects intersect. Geometric intersection graph classes have been studied from both the theoretical and practical point of view. On the one hand, many hard problems can be efficiently solved or approximated when the input graph is restricted to a geometric intersection class of graphs. On the other hand, these graphs appear naturally in many applications such as sensor networks, scheduling problems, and others. Recently, in the context of distributed certification and distributed interactive proofs, the recognition of graph classes has started to be intensively studied. Different results related to the recognition of trees, bipartite graphs, bounded diameter graphs, triangle-free graphs, planar graphs, bounded genus graphs, H-minor free graphs, etc., have been obtained. The goal of the present work is to design efficient distributed protocols for the recognition of relevant geometric intersection graph classes, namely permutation graphs, trapezoid graphs, circle graphs and polygon-circle graphs. More precisely, for the two first classes we give proof labeling schemes recognizing them with logarithmic-sized certificates. For the other two classes, we give three-round distributed interactive protocols that use messages and certificates of size O(log n). Finally, we provide logarithmic lower-bounds on the size of the certificates on the proof labeling schemes for the recognition of any of the aforementioned geometric intersection graph classes.

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“…Meanwhile, polynomial-time algorithms have been studied intensively on some special classes of graphs such as tree graphs [22], weighted tree graphs [3], inflated tree graphs [16], convex bipartite graph [14,20], permutation graphs [6,17,18], strongly chordal graphs [4],…”

Section: Introductionmentioning

confidence: 99%

A Linear-Time Algorithm for the Weighted Paired-Domination Problem on Block Graphs

Lin¹,

Hsieh²

2016

Preprint

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In a graph G = (V, E), a vertex subset S ⊆ V (G) is said to be a dominating set of G if every vertex not in S is adjacent to a vertex in S. A dominating set S of G is called a paired-dominating set of G if the induced subgraph G[S] contains a perfect matching. In this paper, we propose an O(n + m)-time algorithm for the weighted paireddomination problem on block graphs using dynamic programming, which strengthens the results in [

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An O(n)-Time Algorithm for the Paired-Domination Problem on Permutation Graphs

Cited by 13 publications

References 28 publications

Complexity of Paired Domination in AT-free and Planar Graphs

Complexity of Paired Domination in AT-free and Planar Graphs

Distributed Interactive Proofs for the Recognition of Some Geometric Intersection Graph Classes

A Linear-Time Algorithm for the Weighted Paired-Domination Problem on Block Graphs

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