Total Roman Reinforcement in Graphs

• Main Authors: Ahangar H. Abdollahzadeh, Amjadi J., Chellali M., Nazari-Moghaddam S., Sheikholeslami S.M.
• Format: Article
• Language: English
• Published: Sciendo 2019-11-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: total roman domination number; total roman reinforcement number; 05c69
• Online Access: https://doi.org/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.