The complement of the intersection graph of ideals of a poset

Let (P,<=) be an atomic poset with the least element 0. The complement of the intersection graph of ideals of P, denoted by Gamma(P), is defined to be a graph whose vertices are all non-trivial ideals of P and two distinct vertices I and J are adjacent if and only if I boolean AND J = {0}. In this paper, we consider the complement of the intersection graph of ideals of a poset. We prove that Gamma(P) is totally disconnected or diam(Gamma(P) \ (SIC)) is an element of{1, 2, 3}, where (SIC) is the set of all isolated vertices of Gamma(P). We show that gr(Gamma(P)) is an element of{3, 4,infinity}. Also, we characterize all posets whose complement of the intersection graph is forest, unicyclic or complete r-partite graph. Among other results, we prove that Gamma(P) is weakly perfect; and it is perfect if and only if |Atom(P)|<= 4. Finally, we show that Gamma(P) is class 1, where P = Atom(P) boolean OR {0}.