Perfect Italian Domination in Cographs

For a graph G = (V-G, E-G), a perfect Italian dominating function on G is a function g : V-G -> {0, 1, 2} satisfying the condition that for each vertex v with g(v) = 0 , the sum of the function values assigned to the neighbors of v is exactly two, that is, Sigma g(u) = 2 where the sum is taken over all neighbors of v . The weight of g , denoted by w(g) is defined Sigma g(v) where the sum is taken over all v is an element of V-G. The perfect Italian domination number of G , denoted gamma(p)(I) (G) , is the minimum weight of a perfect Italian dominating function of G . In this paper, we prove that the perfect Italian domination number of a connected cograph, a graph containing no induced path on four vertices, belongs to {1, 2, 3, 4} or equals to the order of the cograph. We prove that there is no connected cograph with perfect Italian domination number k , where k is an element of {5, 6, 7, 8, 9}. We also show that for any positive inte-ger k, k is not an element of {5, 6, 7, 8, 9}, there exists a connected cograph whose perfect Italian domination number is k . Moreover, we devise a linear time algorithm that computes the perfect Italian domination number in cographs. (c) 2020 Elsevier Inc. All rights reserved.