Complexity and Algorithms for Semipaired Domination in Graphs

Henning, Michael A.; Pandey, Arti; Tripathi, Vikash

doi:10.1007/s00224-020-09988-3

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…We first show that the MINIMUM SEMIPAIRED DOMINAION problem is APX-complete for graphs with maximum degree 4. To show this result, we prove that the reduction given in the proof of Theorem 2 in [10] is an L-reduction. So, we show an L-reduction from the MINIMUM VERTEX COVER PROBLEM for graphs with maximum degree 3 [1].…”

Section: Apx-completeness For Bounded Degree Graphsmentioning

confidence: 92%

“…Since the SEMIPAIRED DOMINATION DECISION problem is known to be NP-complete for general graphs [10], the following theorem follows directly from Lemma 2.2.…”

Section: Complexity Difference Between Paired Domination and Semipair...mentioning

confidence: 99%

“…In this section, we show that the MINIMUM SEMIPAIRED DOMINAION problem is APX-complete for graphs with maximum degree 3. It is known that the MINIMUM SEMIPAIRED DOMINATION problem for a graph G with maximum degree ∆ can be approximated with an approximation ratio of 1+ln(2∆+2) [10]. Hence the MINIMUM SEMIPAIRED DOMINATION problem is in APX for bounded degree graphs.…”

Section: Apx-completeness For Bounded Degree Graphsmentioning

confidence: 99%

“…A relaxed version of paired domination, known as semipaired domination was introduced by Haynes and Henning [3], and further studied by others [4,8,9,10,14]. For a graph G with no isolated vertex, a semipaired dominating set, abbreviated as semi-PD-set, is a dominating set D of G such that the vertices in D can be partitioned into 2-sets such that if {u, v} is a 2-set, then uv ∈ E(G) or the distance between u and v is 2.…”

Section: Introductionmentioning

confidence: 99%

“…The algorithmic study of the MINIMUM SEMIPAIRED DOMINATION problem was initiated by Henning et al in [10]. They proved that the SEMIPAIRED DOMINATION DECISION problem is NP-complete even for bipartite graphs and chordal graphs.…”

Section: Introductionmentioning

confidence: 99%

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Semipaired Domination in Some Subclasses of Chordal Graphs

Henning,

Pandey,

Tripathi

2020

Preprint

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A dominating set D of a graph G without isolated vertices is called semipaired dominating set if D can be partitioned into 2-element subsets such that the vertices in each set are at distance at most 2. The semipaired domination number, denoted by γ pr2 (G) is the minimum cardinality of a semipaired dominating set of G. Given a graph G with no isolated vertices, the MINIMUM SEMIPAIRED DOM-INATION problem is to find a semipaired dominating set of G of cardinality γ pr2 (G). The decision version of the MINIMUM SEMIPAIRED DOMINATION problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the MINIMUM SEMIPAIRED DOMINATION problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the MINIMUM SEMIPAIRED DOMINATION problem is APX-complete for graphs with maximum degree 3.

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Section: Apx-completeness For Bounded Degree Graphsmentioning

confidence: 92%

“…Since the SEMIPAIRED DOMINATION DECISION problem is known to be NP-complete for general graphs [10], the following theorem follows directly from Lemma 2.2.…”

Section: Complexity Difference Between Paired Domination and Semipair...mentioning

confidence: 99%

Section: Apx-completeness For Bounded Degree Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semipaired Domination in Some Subclasses of Chordal Graphs

Henning,

Pandey,

Tripathi

2020

Preprint

Self Cite

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Comparative Analysis of (1 + λ)Evolutionary Algorithms Based on Comparison-in-time Model

Feng

Zhou

Tan

et al. 2022

Preprint

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Runtime comparative analysis of evolutionary algorithms (EAs) is an important topic in the theoretical research of evolutionary computation. Few studies have been conducted to compare the performance of the runtime analysis. The runtime of EAs can be measured by the first hitting time (FHT) to find the optimal solution. In this paper, we present the design of an equal-in-time model based on the stochastic process model of FHT and the property of equivalence relation, which realizes by expected FHT (EFHT) equivalence class division of difference EAs. With the equal-in-time model, we propose a comparison-in-time model, which can directly compare and analyze the EFHT of different EAs. To verify the feasibility of the proposed comparison-in-time model on a multi-individual population, we conducted a case study to analyze the performance of (1 + λ)EA with different selection operators and different variation operators. Experimental results show that the proposed comparison-in-time model is applicable to the comparison of (1 + λ)EA and is consistent with the comparative analysis results of (1 + 1)EA.

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Semipaired Domination in Some Subclasses of Chordal Graphs

Henning¹,

Pandey²,

Tripathi³

2021

Discrete Mathematics &Amp; Theoretical Computer Science

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A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.

show abstract

Complexity and Algorithms for Semipaired Domination in Graphs

Cited by 4 publications

References 13 publications

Semipaired Domination in Some Subclasses of Chordal Graphs

Semipaired Domination in Some Subclasses of Chordal Graphs

Comparative Analysis of (1 + λ)Evolutionary Algorithms Based on Comparison-in-time Model

Semipaired Domination in Some Subclasses of Chordal Graphs

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