Trees with vertex-edge roman domination number twice the domination number minus one

Kumar, H. Naresh; Venkatakrishnan, Y. B.

doi:10.22199/issn.0717-6279-2020-06-0084

Cited by 5 publications

(1 citation statement)

References 7 publications

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“…Since then, over 100 papers on Roman domination and its variants have been published. Perfect Roman domination [5,6], Italian domination [7,8], perfect Italian domination [9], double-Roman domination [10], perfect double-Roman domination [11], double-Italian domination [12], total Roman domination [13,14], vertex-edge Roman domination [15], and vertex-edge perfect Roman domination [16] are a few examples of Roman domination variants.…”

Section: Introductionmentioning

confidence: 99%

Total Perfect Roman Domination

Almulhim

2023

Symmetry

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A total perfect Roman dominating function (TPRDF) on a graph G=(V,E) is a function f from V to {0,1,2} satisfying (i) every vertex v with f(v)=0 is a neighbor of exactly one vertex u with f(u)=2; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is ∑v∈Vf(v). The total perfect Roman domination number of G, denoted by γtRp(G), is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible γtRp(G). We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related γtRp(G) to perfect domination γp(G) by proving γtRp(G)≤3γp(G) for every graph G, and we characterized trees T of order n≥3 for which γtRp(T)=3γp(T). This notion can be utilized to develop a defensive strategy with some properties.

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Section: Introductionmentioning

confidence: 99%

Total Perfect Roman Domination

Almulhim

2023

Symmetry

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The Vertex-Edge Locating Roman Domination of Some Graphs

Akwu,

Agbaji

2024

Qeios

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In this paper, we introduce the concept of vertex-edge locating Roman dominating functions in graphs. A vertex-edge locating Roman dominating (\(ve-LRD\)) function of a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow\{0,1,2\}\) such that the following conditions are satisfied: (i) for every adjacent vertices \(u,v\) with \(f(u)=0\) or \(f(v)=0\), there exists a vertex \(w\) at distance \(1\) or \(2\) from \(u\) or \(v\) with \(f(w)=2\), (ii) for every edge \(uv\in E\), \(max[f(u),f(v)]\neq 0\) and (iii) any pair of distinct vertices \(u,v\) with \(f(u)=f(v)=0\) does not have a common neighbour \(w\) with \(f(w)=2\) . The weight of \(ve\)-LRD function is the sum of its function values over all the vertices. The vertex-edge locating Roman domination number of \(G\) denoted by \(\gamma_{ve-LR}^P(G)\) is the minimum weight of a \(ve\)-LRD function in \(G\). We proved that the vertex-edge locating Roman domination problem is NP complete for bipartite graphs. Also, we present the upper and lower bonds of \(ve\)-LRD function for trees. Lastly, we give the upper bounds of \(ve\)-LRD function for some connected graphs.

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The Vertex-Edge Locating Roman Domination of Some Graphs

Akwu,

Agbaji

2024

Preprint

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In this paper, we introduce the concept of vertex-edge locating Roman dominating functions in graphs. A vertex-edge locating Roman dominating (\({ve} - {LRD}\)) function of a graph \(G = {(V,E)}\) is a function \(f:{{V{(G)}}\rightarrow{\{ 0,1,2\}}}\) such that the following conditions are satisfied: (i) for every adjacent vertices \(u,v\) with \({f{(u)}} = 0\) or \({f{(v)}} = 0\), there exists a vertex \(w\) at distance \(1\) or \(2\) from \(u\) or \(v\) with \({f{(w)}} = 2\), (ii) for every edge \({uv} \in E\), \({max{\lbrack{f{(u)}},{f{(v)}}\rbrack}} \neq 0\) and (iii) any pair of distinct vertices \(u,v\) with \({f{(u)}} = {f{(v)}} = 0\) does not have a common neighbour \(w\) with \({f{(w)}} = 2\) . The weight of \(ve\)-LRD function is the sum of its function values over all the vertices. The vertex-edge locating Roman domination number of \(G\) denoted by \(\gamma_{{ve} - {LR}}^{P}{(G)}\) is the minimum weight of a \(ve\)-LRD function in \(G\). We proved that the vertex-edge locating Roman domination problem is NP complete for bipartite graphs. Also, we present the upper and lower bonds of \(ve\)-LRD function for trees. Lastly, we give the upper bounds of \(ve\)-LRD function for some connected graphs.

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Trees with vertex-edge roman domination number twice the domination number minus one

Cited by 5 publications

References 7 publications

Total Perfect Roman Domination

Total Perfect Roman Domination

The Vertex-Edge Locating Roman Domination of Some Graphs

The Vertex-Edge Locating Roman Domination of Some Graphs

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