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A graph theoretical model called Roman domination in \linebreak 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 $f : V \to \{0, 1, 2 \}$ 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.~\cite{Ruiz} 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.
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참고문헌 (4)
참고문헌 신청
M. P. Alvarez-Ruiz / 2017 / On the strong Roman domination number of graphs / Discrete Appl. Math 231:44~59
E. J. Cockayne / 2004 / Roman domination in graphs / Discrete Math 278(1-3):11~22
C. S. ReVelle / 2000 / Defendens imperium romanum: a classical problem in military strategy / Amer. Math. Monthly 107(7):585~594
L. Stewart / 1999 / Defend the roman empire! / Sci. Amer 281(6):136~139
E. J. Cockayne / 2004 / Roman domination in graphs / Discrete Math 278(1-3):11~22
C. S. ReVelle / 2000 / Defendens imperium romanum: a classical problem in military strategy / Amer. Math. Monthly 107(7):585~594
L. Stewart / 1999 / Defend the roman empire! / Sci. Amer 281(6):136~139
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