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Perfect Italian domination on planar and regular graphs. (English) Zbl 1450.05066
Let $$G = (V, E)$$ be a graph. A perfect Italian dominating function of a graph $$G$$ is a function $$f : V \rightarrow \{0, 1, 2\}$$ such that for every vertex $$v\in V$$ with $$f(v) = 0$$, it holds that $$\sum_{u\in N(v)} f (u) = 2$$. The perfect Italian domination number of $$G$$, denoted by $$\gamma_I^p(G)$$, is the minimum weight of any perfect Italian dominating function of $$G$$. For $$k \ge 1$$, a $$k$$-fair dominating set of $$G$$ is a dominating set $$D$$ such that $$|N(v)\cap D| = k$$ for every $$v \in V \setminus D$$. The $$k$$-fair domination number of $$G$$, denoted by $$fd_k(G)$$, is the minimum cardinality of a $$k$$-fair dominating set in $$G$$. For a graph $$G$$, let $$\overline{G}$$ be the complement of $$G$$.
In this paper, there are many interesting results and some of main of them are stated as follows.
Theorem 1. For every graph $$G$$, it holds that $$\gamma_I^p(G) \le fd_2(G)$$.
Theorem 2. A connected graph $$G$$ with $$\gamma_I^p(G)> 2$$ has $$\gamma_I^p(G) = 3$$ if and only if $$G$$ has a $$2$$-fair dominating set $$D$$ of size $$3$$.
Theorem 3. A connected graph $$G$$ with $$\gamma_I^p(G)> 2$$ has $$\gamma_I^p(G) = 3$$ if and only if $$\overline{G}$$ has a perfect dominating set of size $$3$$.
Theorem 4. There is an infinite family of $$n$$-vertex connected planar graphs $$G$$ such that $$\gamma_I^p(G) = n$$.
Theorem 5. A connected graph $$G$$ on $$n$$ vertices with maximum degree $$\Delta$$ has $$\gamma_I^p(G)\ge 2n/(\Delta + 2)$$.
Theorem 6. Every connected cubic graph $$G$$ with $$n$$ vertices has $$\frac{2}{5}n \le \gamma_I^p(G)\le \frac{2}{3}n$$. Moreover, these bounds are tight.
Theorem 7. Perfect Italian domination is NP-complete for bipartite planar graphs.
##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
Italian domination, perfect Italian
##### Software:
GENREG; House of Graphs
Full Text:
##### References:
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