Algorithmic aspects of b-disjunctive domination in graphs

Panda, B. S.; Pandey, Arti; Shrivastava, Paul

doi:10.1007/s10878-017-0112-6

Cited by 14 publications

(5 citation statements)

References 16 publications

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“…In particular, there exist graph classes for which the first problem is polynomial-time solvable whereas the second problem is NP-complete and vice versa. Similar study has been made between domination and other domination parameters in [14,20,21]. The DOMINATION problem is linear time solvable for doubly chordal graphs [3], but the SCDM problem is NP-complete for this class of graphs which is proved in section 2.1.…”

Section: Complexity Difference In Domination and Secure Connected Dom...mentioning

confidence: 66%

Algorithmic aspects of secure connected domination in graphs

Kumar¹,

Reddy²

2021

Discuss. Math. Graph Theory

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A dominating set S is an Isolate Dominating Set (IDS ) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S ⊆ V is an isolate secure dominating set (ISDS ), if for each vertex u ∈ V \S, there exists a neighboring vertex v of u in S such that (S \ {v}) ∪ {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by γ 0s (G). Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.

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Section: Complexity Difference In Domination and Secure Connected Dom...mentioning

confidence: 66%

Algorithmic aspects of secure connected domination in graphs

Kumar¹,

Reddy²

2021

Discuss. Math. Graph Theory

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“…In particular, there exist graph classes for which the decision version of the first problem is polynomial-time solvable whereas the decision version of the second problem is NP-complete and vice versa. Similar study has been made between domination and other domination parameters in [13,16]. either v i or a i , for each i, where 1 ≤ i ≤ n must be included in every InSDS of G , and hence |S| ≥ 2n.…”

Section: Complexity Difference In Domination and Independent Secure Dominationmentioning

confidence: 77%

Algorithmic Aspects of Some Variants of Domination in Graphs

Kumar

Reddy

2020

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V . A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v}) ∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γis(G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5.

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“…In particular, there exist graph classes for which the decision version of the first problem is polynomial-time solvable whereas the second problem is NP-complete and vice versa. Similar study has been made between domination and other domination parameters in [18][19][20]. We construct a new class of graphs in which the MR3DP can be solved trivially, whereas the decision version of the DOMINATION problem is NP-complete, which is defined as follows.…”

Section: Complexity Contrast Between Domination and Roman {3}-dominat...mentioning

confidence: 96%

Algorithmic aspects of Roman domination in graphs

Padamutham

Palagiri

2020

J. Appl. Math. Comput.

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Let G be a simple, undirected graph. A function g : V (G) → {0, 1, 2, 3} having the property that v∈N G (u) g(v) ≥ 3, if g(u) = 0, and v∈N G (u) g(v) ≥ 2, if g(u) = 1 for any vertex u ∈ G, where NG(u) is the set of vertices adjacent to u in G, is called a Roman {3}-dominating function (R3DF) of G. The weight of a R3DF g is the sum g(V ) = v∈V g(v). The minimum weight of a R3DF is called the Roman {3}-domination number and is denoted by γ {R3} (G). Given a graph G and a positive integer k, the Roman {3}-domination problem (R3DP) is to check whether G has a R3DF of weight at most k. In this paper, first we show that the R3DP is NP-complete for chordal graphs, planar graphs and for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. The minimum Roman {3}-domination problem (MR3DP) is to find a R3DF of minimum weight in the input graph. We show that MR3DP is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. We propose a 3(1 + ln(∆ − 1))-approximation algorithm for the MR3DP, where ∆ is the maximum degree of G and show that the MR3DP problem cannot be approximated within (1 − ϵ) ln |V | for any ϵ > 0 unless N P ⊆ DT IM E(|V | O(log log |V |) ). Next, we show that the MR3DP problem is APX-complete for graphs with maximum degree 4. We also show that the domination and Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, an ILP formulation for MR3DP is proposed.

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Algorithmic aspects of b-disjunctive domination in graphs

Cited by 14 publications

References 16 publications

Algorithmic aspects of secure connected domination in graphs

Algorithmic aspects of secure connected domination in graphs

Algorithmic Aspects of Some Variants of Domination in Graphs

Algorithmic aspects of Roman domination in graphs

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