DOI QR코드

DOI QR Code

TREES WITH EQUAL STRONG ROMAN DOMINATION NUMBER AND ROMAN DOMINATION NUMBER

  • Chen, Xue-Gang (Department of Mathematics North China Electric Power University) ;
  • Sohn, Moo Young (Department of Mathematics Changwon National University)
  • 투고 : 2018.01.17
  • 심사 : 2018.10.11
  • 발행 : 2019.01.31

초록

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.

키워드

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)

참고문헌

  1. 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
  2. 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
  3. 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
  4. L. Stewart, Defend the roman empire!, Sci. Amer. 281 (1999), no. 6, 136-139. https://doi.org/10.1038/scientificamerican1299-136