Perfect Italian domination on planar and regular graphs.

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 2. 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 γIp(G), is the minimum weight of any perfect Italian dominating function of G. While introducing the parameter, Haynes and Henning (2019) also proposed the problem of determining the best possible constants cG such that γIp(G)≤cG×n for all graphs of order n when G is in a particular class G of graphs. They proved that cG=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 cG=1 and for cubic graphs by proving that cG=2∕3. For split graphs, we also show that cG=1. In addition, we characterize the graphs G with γIp(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.