Perfect Italian domination on planar and regular graphs
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A perfect Italian dominating function of a graph G=(V,E) is a function f : V →{0,1,2} such that for every vertex f(v) = 0, it holds that ∑_u ∈ 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_G such that γ^p_I(G) ≤ c_G× n for all graphs of order n when G is in a particular class G of graphs. They proved that c_G = 1 when 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_G = 1 and for cubic graphs by proving that c_G = 2/3. For split graphs, we also show that c_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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