Total Roman reinforcement in graphs

• Published in: Discussiones Mathematicae. Graph Theory Vol. 39; no. 4; pp. 787 - 803
• Main Authors: Ahangar, H. Abdollahzadeh, Amjadi, Jafar, Chellali, Mustapha, Nazari-Moghaddam, S., Sheikholeslami, Seyed Mahmoud
• Format: Journal Article
• Language: English
• Published: University of Zielona Góra 01.11.2019
• Subjects: 05c69; total roman domination number; total roman reinforcement number
• ISSN: 1234-3099; 2083-5892
• DOI: 10.7151/dmgt.2108

A total Roman dominating function on a graph G is a labeling f : V (G) → {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2 and the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The minimum weight of a total Roman dominating function on a graph G is called the total Roman domination number of G. The total Roman reinforcement number rtR (G) of a graph G is the minimum number of edges that must be added to G in order to decrease the total Roman domination number. In this paper, we investigate the proper- ties of total Roman reinforcement number in graphs, and we present some sharp bounds for rtR (G). Moreover, we show that the decision problem for total Roman reinforcement is NP-hard for bipartite graphs.