Abstract
In this paper, we derive sharp upper and lower bounds on the sum and product , where is the complement of graph G. We also show that for each tree T of order n ≥ 2, γ[3R](T) ≤ 3n + s(T)/2 and γ[3R](T) ≥ ⌈4(n(T) + 2 − ℓ(T))/3⌉, where s(T) and ℓ(T) are the number of support vertices and leaves of T.
Highlights
E distance d(u, v) between two vertices u and v of a graph G is the length of a shortest (u, v)-path in G. e maximum distance among all pairs of vertices in G is the diameter of G, which is denoted by diam (G)
E Roman domination in graphs is well studied in graph theory. e topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. e deployment of the legions around the Empire is designed in such a way that a sudden attack to any undefended city could be quelled by a legion from a strong neighbor. ere is an additional condition: no legion can move and doing so leaves its base city defenceless
We first present Nordhaus–Gaddumtype bounds for the triple Roman domination number
Summary
E distance d(u, v) between two vertices u and v of a (connected) graph G is the length of a shortest (u, v)-path in G. e maximum distance among all pairs of vertices in G is the diameter of G, which is denoted by diam (G). For a vertex v in a rooted tree T, the maximal subtree at v is the subtree of T induced by v and its descendants and is denoted by Tv. In the field of chemistry, graph theory has provided many useful tools, such as topological indices. Topological indices and domination in graphs are the essential topics in the theory of graphs. We continue the study of a variant of Roman domination in graphs: the triple Roman domination. We determine various bounds on the triple Roman domination number for general graphs, and we give exact values for some graph families. We first present Nordhaus–Gaddumtype bounds for the triple Roman domination number. If G is a connected graph on n vertices, c[3R](G) 6 if and only if there are two nonjoin vertices in V(G) with degree Δ(G) n − 2
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