论文标题
意大利统治和完美的意大利统治在Sierpinski图上
Italian Domination and Perfect Italian Domination on Sierpinski Graphs
论文作者
论文摘要
图G的意大利主导函数（IDF）是一个函数$ f：v（g）\ rightarrow \ {0,1,2 \} $，可满足每一个$ v $ in v $ in $ f（v）= 0 $，$ \ sum_ sum_ sum_ {u \ in n（v）in（v）f（v）f（v）f（u）ins y y y y y y d y d p a y d y d p a n v $的条件。 \ sum_ {v \ in V} f（v）$和意大利统治号码，$γ_i（g）$，是IDF的最小重量。 IDF是$ g $的完美意大利主导功能（PID），如果对于$ f（g）$中的每个顶点$ v \，$ f（v）= 0 $ $ f $ $ f $分配给$ v $的邻居的总权重恰好是2，即$ u $的所有邻居都为$ f $ f $ f $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v $ v（verte $ v） $ v $和$ w $，$ f（v）= f（w）= 1 $。 PID函数的重量为$ f（v）= \ sum_ {u \ in V（g）} f（u）$。 $ g $的完美意大利统治数，由$γ^{p} _ {i}（g）表示，$是$ g $的PID功能的最小重量。在本文中，我们获得了意大利的统治号码和Sierpiński图的完美意大利统治数。
An Italian dominating function (IDF) of a graph G is a function $ f: V(G) \rightarrow \{0,1,2\} $ satisfying the condition that for every $ v\in V $ with $ f(v) = 0$, $\sum_{ u\in N(v)} f(u) \geq 2. $ The weight of an IDF on $G$ is the sum $ f(V)= \sum_{v\in V}f(v) $ and the Italian domination number, $ γ_I(G) $, is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on $G$, if for every vertex $ v \in V(G) $ with $ f(v) = 0 $ the total weight assigned by $f$ to the neighbours of $ v $ is exactly 2, i.e., all the neighbours of $u$ are assigned the weight 0 by $f$ except for exactly one vertex $v$ for which $ f(v) = 2 $ or for exactly two vertices $v$ and $w$ for which $ f(v) = f(w) = 1 $. The weight of a PID- function is $f(V)=\sum_{u \in V(G)}f(u)$. The perfect Italian domination number of $G$, denoted by $ γ^{p}_{I}(G), $ is the minimum weight of a PID-function of $G$. In this paper we obtain the Italian domination number and perfect Italian domination number of Sierpiński graphs.
