On recognition of threshold tolerance graphs and their complements

A graph G = (V,E) is a threshold tolerance graph if each vertex v ∈ V can be assigned a weight wv and a tolerance tv such that two vertices x, y ∈ V are adjacent if wx +wy ≥ min(tx, ty). Currently, the most efficient recognition algorithm for threshold tolerance graphs is the algorithm of Monma, Reed, and Trotter which has an O(n) runtime. We give an O(n) algorithm for recognizing threshold tolerance and their complements, the threshold tolerance (co-TT) graphs, resolving an open question of Golumbic, Weingarten, and Limouzy.