Fast Algorithms for Independent Domination and Efficient Domination in Trapezoid Graphs

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2 +P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3.

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“…Assume that for z ∈ N 2 , zd ∈ E with d ∈ D ∩ Q 1 . Then by (24) and the e.d. property, z has a non-neighbor in every Q i , i ≥ 2.…”

Section: A Direct Solution For the Wed Problem On P 5 -Free Graphsmentioning

confidence: 94%

New Polynomial Cases of the Weighted Efficient Domination Problem

Brandstädt

Milanič

Nevries

2013

Mathematical Foundations of Computer Science 2013

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“…Consequently, the Min-WED problem is NP-complete in every class of graphs where the ED problem is NP-complete. On the other hand, the Min-WED problem is solvable in polynomial time for trees [40], cocomparability graphs [10,14], split graphs [12], interval graphs [13,14], circular-arc graphs [13], permutation graphs [28], trapezoid graphs [28,29], bipartite permutation graphs [32], convex bipartite graphs [7] and their superclass interval bigraphs [7], distance-hereditary graphs [32], block graphs [41] and hereditary efficiently dominatable graphs [16,34]. Since negative weights are allowed, the Max-WED problem is equivalent to the Min-WED problem.…”

Section: Minimum Weight Efficient Dominating Set (Min-wed)mentioning

confidence: 99%

“…The inclusion relations in the figure were verified with help of the Information System on Graph Classes and their Inclusions [21]. NP-c [5] O(n + m) [12] O(n + m) [41] planar cubic bipartite O(n) * [32] O(n 2 ) [10,14] O(n + m) [13,14] O(n(n + m)) [13] NP-c permutation O(n) [40] O(n + m) * [28] O(n) * [32] O(n log n) * [29] O(n log log n + m) * [28] O(m + n)…”

Section: Minimum Weight Efficient Dominating Set (Min-wed)mentioning

confidence: 99%

“…The ED problem is NP-complete even for restricted graph classes such as planar cubic graphs [25], bipartite graphs [41], planar bipartite graphs [32], chordal bipartite graphs [32], chordal graphs [41], and line graphs of planar bipartite graphs of maximum degree three [5]. On the other hand, the ED problem is polynomial for several graph classes, including trees [1,17], block graphs [41], interval graphs [13,14,24,26], circular-arc graphs [13,24], cocomparability graphs [10,14], bipartite permutation graphs [32], permutation graphs [28], distance-hereditary graphs [32], trapezoid graphs [28,29], split graphs [12], dually chordal graphs [7], AT-free graphs [7,8], and hereditary efficiently dominatable graphs [34]. The efficient domination problem has also been studied from a parameterized point of view, see, e.g., [9,23].…”

Section: Introductionmentioning

confidence: 99%

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Polynomial-time algorithms for weighted efficient domination problems in AT-free graphs and dually chordal graphs

Brandstädt

Fičur

Leitert

et al. 2015

Information Processing Letters

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An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs.

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“…For trapezoid graphs, Liang [10] showed that the minimum weighted domination problem and the total domination problem can be solved in OðmnÞ time. Lin [12] showed that the minimum weighted independent domination in trapezoid graphs can be found in Oðn log nÞ time. Srinivasan et al [15] showed that the minimum weighted connected domination problem in trapezoid graphs can be solved in Oðm þ n log nÞ time.…”

Section: Introductionmentioning

confidence: 99%