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Total Roman {3}-domination in Graphs

1
Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China
2
Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran
3
Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(2), 268; https://doi.org/10.3390/sym12020268
Received: 4 January 2020 / Revised: 2 February 2020 / Accepted: 3 February 2020 / Published: 9 February 2020
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.
Keywords: Roman domination; Roman {3}-domination; Total Roman {3}-domination Roman domination; Roman {3}-domination; Total Roman {3}-domination
MDPI and ACS Style

Shao, Z.; Mojdeh, D.A.; Volkmann, L. Total Roman {3}-domination in Graphs. Symmetry 2020, 12, 268.

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