On the complexity of signed and minus total domination in graphs

Lee, Chuan-Min

doi:10.1016/j.ipl.2009.08.002

Cited by 7 publications

(6 citation statements)

References 19 publications

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“…Then we adopt a unified method and show that the signed (minus) total domination problem can be solved in O(n + m) time for a chordal bipartite graph G if a weak elimination ordering of G is provided. This improves the complexity of finding the minimum weight of signed (minus) total domination function for chordal bipartite graphs proposed in [15] from O(n 2 ) time to O(n + m) time.…”

Section: Complexity Of Certain Functional Variants Of Total Dominatiomentioning

confidence: 88%

“…minus) total dominating function in chordal bipartite graphs. For this, we introduce R-total domination in graphs which is also introduced in [15]. Let , d, I 1 be fixed integers and , d > 0.…”

Section: Signed and Minus Total Domination In Chordal Bipartite Graphsmentioning

confidence: 99%

“…and an integer l > 0, in Min-Signed-Total-Zero-Dom-Function problem, it is required to decide whether G has a signed total dominating function of weight at most l. The following theorem is taken from [15] which will be used later.…”

Section: Complexity Of Certain Functional Variants Of Total Dominatiomentioning

confidence: 99%

“…Min-Minus-Total-Dom-Function) problem, it is required to decide whether G has a signed (resp. minus) total dominating function of G of size at most l. The literature on signed (minus) total domination function can be found in [12,14,15,18,19,[22][23][24].…”

Section: Introductionmentioning

confidence: 99%

“…[15]). Suppose that G = (X, Y, E) is a bipartite graph with a labeling function R as mentioned above.Let f X (resp.…”

mentioning

confidence: 99%

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Complexity of Certain Functional Variants of Total Domination in Chordal Bipartite Graphs

Pradhan

2012

Discrete Math. Algorithm. Appl.

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In this paper, we consider minimum total domination problem along with two of its variations namely, minimum signed total domination problem and minimum minus total domination problem for chordal bipartite graphs. In the minimum total domination problem, the objective is to find a smallest size subset TD ⊆ V of a given graph G = (V, E) such that |TD∩NG(v)| ≥ 1 for every v ∈ V. In the minimum signed (minus) total domination problem for a graph G = (V, E), it is required to find a function f : V → {-1, 1} ({-1, 0, 1}) such that f(NG(v)) = ∑u∈NG(v)f(u) ≥ 1 for each v ∈ V, and the cost f(V) = ∑v∈V f(v) is minimized. We first show that for a given chordal bipartite graph G = (V, E) with a weak elimination ordering, a minimum total dominating set can be computed in O(n + m) time, where n = |V| and m = |E|. This improves the complexity of the minimum total domination problem for chordal bipartite graphs from O(n2) time to O(n + m) time. We then adopt a unified approach to solve the minimum signed (minus) total domination problem for chordal bipartite graphs in O(n + m) time. The method is also able to solve the minimum k-tuple total domination problem for chordal bipartite graphs in O(n + m) time. For a fixed integer k ≥ 1 and a graph G = (V, E), the minimum k-tuple total domination problem is to find a smallest subset TDk ⊆ V such that |TDk ∩ NG(v)| ≥ k for every v ∈ V.

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Section: Complexity Of Certain Functional Variants Of Total Dominatiomentioning

confidence: 88%

Section: Signed and Minus Total Domination In Chordal Bipartite Graphsmentioning

confidence: 99%

Section: Complexity Of Certain Functional Variants Of Total Dominatiomentioning

confidence: 99%