平面图和正则图的完全意大利控制
Perfect Italian domination on planar and regular graphs
摘要：
图$G=(V,E)$的完美意大利控制函数是函数$f : V \to \{0,1,2\}$，使得对于每个顶点$f(v) = 0$，它保持$\sum_{u \in N(v)} f(u) = 2$，即，由$f$分配给$v$的邻居的标签的权重正好是2。一个完美意大利函数的权是顶点权的和。$G$的完美意大利控制数，记作$γ^p_I(G)$，是$G$的任何完美意大利控制函数的最小权。在引入参数的同时，Haynes和Henning（离散应用数学（2019），164- 177）还提出了确定最佳可能常数$c_\mathcal{G}$的问题，使得当$G$在特定的图类$\mathcal{G}$中时，对于所有$n$阶的图，$γ^p_I(G) \leq c_\mathcal{G} \times n$。他们证明了当$\mathcal{G}$是二部图类时，$c_\mathcal{G} = 1$是二部图，并提出了平面图和正则图的问题。我们解决了他们的问题，正是平面图证明$c_\mathcal{G} = 1$和立方图证明$c_\mathcal{G} = 2/3$。对于分裂图，我们还表明，$c_\mathcal{G} = 1$。此外，我们还刻画了$γ^p_I(G)$等于2和3的图$G$，并确定了几个简单结构图的参数的精确值。我们的结论是证明它是NP-完全的决定是否一个给定的二部平面图承认一个完美的意大利控制功能的重量$k$。
英文摘要：
A perfect Italian dominating function of a graph $G=(V,E)$ is a function $f : V \to \{0,1,2\}$ such that for every vertex $f(v) = 0$, it holds that $\sum_{u \in N(v)} f(u) = 2$, i.e., the weight of the labels assigned by $f$ to the neighbors of $v$ is exactly two. The weight of a perfect Italian function is the sum of the weights of the vertices. The perfect Italian domination number of $G$, denoted by $γ^p_I(G)$, is the minimum weight of any perfect Italian dominating function of $G$. While introducing the parameter, Haynes and Henning (Discrete Appl. Math. (2019), 164--177) also proposed the problem of determining the best possible constants $c_\mathcal{G}$ such that $γ^p_I(G) \leq c_\mathcal{G} \times n$ for all graphs of order $n$ when $G$ is in a particular class $\mathcal{G}$ of graphs. They proved that $c_\mathcal{G} = 1$ when $\mathcal{G}$ is the class of bipartite graphs, and raised the question for planar graphs and regular graphs. We settle their question precisely for planar graphs by proving that $c_\mathcal{G} = 1$ and for cubic graphs by proving that $c_\mathcal{G} = 2/3$. For split graphs, we also show that $c_\mathcal{G} = 1$. In addition, we characterize the graphs $G$ with $γ^p_I(G)$ equal to 2 and 3 and determine the exact value of the parameter for several simple structured graphs. We conclude by proving that it is NP-complete to decide whether a given bipartite planar graph admits a perfect Italian dominating function of weight $k$.
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