Total Roman {3}-domination in Graphs

Abstract

For a graph G = ( V , E ) with vertex set V = V ( G ) and edge set E = E ( G ) , a Roman { 3 } -dominating function (R { 3 } -DF) is a function f : V ( G ) → { 0 , 1 , 2 , 3 } having the property that ∑ u ∈ N G ( v ) f ( u ) ≥ 3 , if f ( v ) = 0 , and ∑ u ∈ N G ( v ) f ( u ) ≥ 2 , if f ( v ) = 1 for any vertex v ∈ V ( G ) . The weight of a Roman { 3 } -dominating function f is the sum f ( V ) = ∑ v ∈ V ( G ) f ( v ) and the minimum weight of a Roman { 3 } -dominating function on G is the Roman { 3 } -domination number of G, denoted by γ { R 3 } ( G ) . Let G be a graph with no isolated vertices. The total Roman { 3 } -dominating function on G is an R { 3 } -DF f on G with the additional property that every vertex v ∈ V with f ( v ) ≠ 0 has a neighbor w with f ( w ) ≠ 0 . The minimum weight of a total Roman { 3 } -dominating function on G, is called the total Roman { 3 } -domination number denoted by γ t { R 3 } ( G ) . We initiate the study of total Roman { 3 } -domination and show its relationship to other domination parameters. We present an upper bound on the total Roman { 3 } -domination number of a connected graph G in terms of the order of G and characterize the graphs attaining this bound. Finally, we investigate the complexity of total Roman { 3 } -domination for bipartite graphs.

Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

“…The total Roman {3}-domination [24] was studied recently . The total Roman {3}-dominating function(TR3DF) on a graph G is an R{3}DF on G with the additional property that every vertex v ∈ V(G) with f (v) = 0 has a neighbor w with f (w) = 0.…”

“…They also presented an upper bound on the total Roman {3}-domination number of a connected graph G and characterized the graphs arriving this bound. Finally, they investigated that total Roman {3}-domination problem is NP-complete for bipartite graphs [24].…”

“…In this paper, we further investigate the complexity of total Roman {3}-domination in planar graphs and chordal bipartite graphs. Moreover, we give a linear-time algorithm to compute the γ t{R3} for trees which answer the problem that it is possible to construct a polynomial algorithm for computing the number of total Roman {3}-domination for trees [24].…”

Double Roman and double Italian domination

“…The weight of a DIDF f is the sum w(f ) = v∈V (G) f (v), and the minimum weight of a DIDF in a graph G is the double Italian domination number, denoted by γ dI (G). This concept was further studied in [3,4,17].…”

Double Roman Graphs in P(3k, k)

“…It is known that the decision problem associated with γ dR (G) is NP-complete for bipartite and chordal graphs, undirected path graphs, chordal bipartite graphs, and circle graphs [15][16][17]. Closely related problems to double Roman domination were studied in [18][19][20].…”