The intersection graph of ideals of a poset

Abstract

Let (P, ≤) be a partially ordered set (poset, briefly) with a least element 0. The intersection graph of ideal of P, denoted by G(P), is 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 ∩ J ≠ {0}. In this paper, we study some relations between the algebraic properties of posets and graph-theoretic properties of G(P). We investigate the connectivity, diameter, girth and planarity of the intersection graph. Also, among the other things, we show that if the clique number of G(P) is finite, then P is finite too.