A graph theoretical model called Roman domination in graphs originates from the historical background that any undefended place (with no legions) of the Roman Empire must be protected by a stronger neighbor place (having two legions). It is applicable to military and commercial decision-making problems. A Roman dominating function for a graph G = (V, E) is a function $f:V{\rightarrow}\{0,1,2\}$ such that every vertex v with f(v)=0 has at least a neighbor w in G for which f(w)=2. The Roman domination number of a graph is the minimum weight ${\sum}_{v{\in}V}\;f(v)$ of a Roman dominating function. In order to deal a problem of a Roman domination-type defensive strategy under multiple simultaneous attacks, ${\acute{A}}lvarez$-Ruiz et al. [1] initiated the study of a new parameter related to Roman dominating function, which is called strong Roman domination. ${\acute{A}}lvarez$-Ruiz et al. posed the following problem: Characterize the graphs G with equal strong Roman domination number and Roman domination number. In this paper, we construct a family of trees. We prove that for a tree, its strong Roman dominance number and Roman dominance number are equal if and only if the tree belongs to this family of trees.
연구 과제 주관 기관 : National Research Foundation of Korea(NRF)
참고문헌
M. P. Alvarez-Ruiz, T. Mediavilla-Gradolph, and S. M. Sheikholeslami, On the strong Roman domination number of graphs, Discrete Appl. Math. 231 (2017), 44-59.https://doi.org/10.1016/j.dam.2016.12.013
E. J. Cockayne, P. M. Dreyer Sr., S. M. Hedetniemi, and S. T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004), no. 1-3, 11-22.https://doi.org/10.1016/j.disc.2003.06.004
C. S. ReVelle and K. E. Rosing, Defendens imperium romanum: a classical problem in military strategy, Amer. Math. Monthly 107 (2000), no. 7, 585-594.https://doi.org/10.1080/00029890.2000.12005243