特殊图的双罗马控制数的研究

Research on the Double Roman Domination Number of Some Special Graphs

摘要: 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：P_m☒P_n (m = 2, 3)，P_n,t，K_n∗，M (C_n)，M (P_n) 的双罗马控制数的确切值。

Abstract: 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  P_m☒P_n (m = 2, 3),  P_n,t,  K_n∗,  M (C_n),  M (P_n).