TY - GEN

T1 - Complexity of Paired Domination in AT-free and Planar Graphs

AU - Tripathi, Vikash

AU - Kloks, Ton

AU - Pandey, Arti

AU - Paul, Kaustav

AU - Wang, Hung Lung

N1 - Publisher Copyright:
© 2022, Springer Nature Switzerland AG.

PY - 2022

Y1 - 2022

N2 - 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 NP-complete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput. Sci., 591 (2015 ) : 99 - 105 and Algorithmica, 82 (2020 ) : 2809 - 2840 ).

AB - 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 NP-complete for planar graphs, which answers an open question asked by Lin et al. (in Theor. Comput. Sci., 591 (2015 ) : 99 - 105 and Algorithmica, 82 (2020 ) : 2809 - 2840 ).

KW - AT-free graphs

KW - Approximation algorithm

KW - Domination

KW - Graph algorithms

KW - NP-completeness

KW - Paired domination

KW - Planar graphs

UR - http://www.scopus.com/inward/record.url?scp=85124667753&partnerID=8YFLogxK

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U2 - 10.1007/978-3-030-95018-7_6

DO - 10.1007/978-3-030-95018-7_6

M3 - Conference contribution

AN - SCOPUS:85124667753

SN - 9783030950170

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 65

EP - 77

BT - Algorithms and Discrete Applied Mathematics - 8th International Conference, CALDAM 2022, Proceedings

A2 - Balachandran, Niranjan

A2 - Inkulu, R.

PB - Springer Science and Business Media Deutschland GmbH

T2 - 8th International Conference on Algorithms and Discrete Applied Mathematics, CALDAM 2022

Y2 - 10 February 2022 through 12 February 2022

ER -