A Note on Threshold Dimension of Permutation Graphs

A graph G(V, E) is a threshold graph if there exist non-negative reals wv, v ∈ V and t such that for every U ⊆ V , P v∈U wv ≤ t if and only if U is a stable set. The threshold dimension of a graph G(V,E), denoted as t(G), is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A permutation graph is a graph that can be represented as the intersection graph of a family of line segments that connect two parallel lines in the Euclidean plane. In this paper we will show that if G is a permutation graph then t(G) ≤ α(G) (where α(G) is the cardinality of maximum independent set in G) and this bound is tight. As a corollary we will show that t(G) ≤ n 2 where n is the number of vertices in the permutation graph G. This bound is also tight.