Discrete Mathematics, Algorithms and Applications, Aug 18, 2021
Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman do... more Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.
Discrete Mathematics, Algorithms and Applications, 2021
Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman do... more Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2020
A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ... more 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 indep...
Discrete Mathematics, Algorithms and Applications, Aug 18, 2021
Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman do... more Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.
Discrete Mathematics, Algorithms and Applications, 2021
Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman do... more Let [Formula: see text] be a connected graph. A function [Formula: see text] is called a Roman dominating function if every vertex [Formula: see text] with [Formula: see text] is adjacent to a vertex [Formula: see text] with [Formula: see text]. If further the set [Formula: see text] is an independent set, then [Formula: see text] is called an outer independent Roman dominating function (OIRDF). Let [Formula: see text] and [Formula: see text]. Then [Formula: see text] is called the outer independent Roman domination number of [Formula: see text]. In this paper, we prove that the decision problem for [Formula: see text] is NP-complete for chordal graphs. We also show that [Formula: see text] is linear time solvable for threshold graphs and bounded tree width graphs. Moreover, we show that the domination and outer independent Roman domination problems are not equivalent in computational complexity aspects.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2020
A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ... more 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 indep...
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Papers by Jakkepalli Pavan Kumar