特殊图的双罗马控制数的研究

Research on the Double Roman Domination Number of Some Special Graphs

令 G = (V (G), E(G)) 是—个简单连通图,函数 f : V (G) → {0, 1, 2, 3} 满足:1) 如果 f (v) = 0,那么至少存在v 的两个邻点 v1, v2, 使得f (v1) = f (v2) = 2,或至少存在 — 个邻点 u 使得f (u) = 3;2) 如果 f (v) = 1,那么至少存在 v 的—个邻点 u 使得f (u) = 2或3。则称 f 为图 G 的—个双罗马控制函数(DRDF)。—个双罗马控制函数的权值为 f (V (G)) = ∑u∈V (G) f (u)。图 G 的双罗马控制函数的最小权值称为图 G 的双罗马控制数,记作 γdR(G)。权值为 γdR(G) 的双罗马控制函数称为 G 的 γdR - 函数。本文主要给出了一些特殊图如:Pm☒Pn (m = 2, 3),Pn,t,Kn∗,M (Cn),M (Pn) 的双罗马控制数的确切值。

Let G = (V (G), E(G)) be a simple connected graph, a function f : V (G) → {0, 1, 2, 3} satisfies with the property that 1) if f (v) = 0, then vertex v must exist at least two neighbors v1, v2 such that f (v1) = f (v2) = 2 or one neighbor u such that f (u) = 3; 2) if f (v) = 1, then there must exist at least one neighbor u of v such that f (u) = 2 or 3, and f is called a double Roman domination function (DRDF). The weight of a DRDF is f (V (G)) = ∑u∈V (G) f (u). The minimum weight of a DRDF on G is the double Roman domination number, denoted by γdR(G). A double Roman domination function with the weight of γdR(G) is called a γdR-function of G. In this paper, we present the exact values of the double Roman domination numbers of some special graphs, such as Pm☒Pn (m = 2, 3), Pn,t, Kn∗, M (Cn), M (Pn).