Recognizing Threshold Tolerance Graphs in O(n 2 ) Time Vincius Fernandes dos Santos ⋆ , Petr A. Golovach ⋆⋆ , Pinar Heggernes ⋆⋆⋆ , Nathan Lindzey † , Ross M. McConnell ‡ , Jeremy P. Spinrad § , Jayme Luiz Szwarcfiter ¶ Abstract. A graph G =(V,E) is threshold tolerance if each vertex v ∈ V can be assigned a weight wv and a tolerance tv such that two vertices x, y ∈ V are adjacent precisely when the sum of their weights exceeds either of their tolerances, that is, 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 4 ) runtime. We give an O(n 2 ) algorithm for recognizing threshold tolerance and complement threshold tolerance (co-TT) graphs, resolving an open question of Golumbic, Weingarten, and Limouzy. 1 Introduction Tolerance graphs are an important subclass of perfect graphs that generalize both interval graphs and permutation graphs [8]. They have been written about extensively and they model constraints in various combinatorial optimization and decision problems [8, 9, 11]. They have a rich structure and history, and interesting relationships to other graph classes. For a detailed overview of the class, see [11]. In this work we limit our attention to threshold tolerance graphs, a subclass of tolerance graphs introduced by Monma, Reed, and Trotter. A graph G =(V,E) is threshold tolerance if each vertex v ∈ V can be assigned a weight w v and a tolerance t v such that two vertices x, y ∈ V are adjacent precisely when the sum of their weights exceeds either of their tolerances, that is, w x + w y ≥ min(t x ,t y ) [13]. When the tolerances of the vertices are all the same, we obtain the threshold graphs [4]. Their complements, the co-threshold tolerance graphs (co-TT graphs ), have also received attention as they have an interesting interpretation as a generalization of interval graphs. A graph G =(V,E) is an interval graph if and only if each vertex v ∈ V can be assigned an interval I v =[a(v),b(v)] on the real line such that two vertices x, y ∈ V are adjacent exactly when their corresponding intervals intersect, in which case I = {[a(v),b(v)] : v ∈ V } forms an interval model of G. See [6, 15, 3] for surveys of the properties of this class and its relationship to other graph classes. To illustrate the relationship of the interval graphs to the threshold graphs, we rephrase the definition: Definition 1. A graph G =(V,E) is an interval graph if and only if there exist functions a, b : V → R such that: ⋆ vinicius@ime.uerj.br, Departamento de Matematica Aplicada, Universidade do Estado do Rio de Janeiro - UERJ, Brasil ⋆⋆ Petr.Golovach@ii.uib.no, Department of Informatics, University of Bergen, P.O.Box 7803, N-5020 Bergen, Norway. ⋆⋆⋆ pinar@ii.uib.no, Department of Informatics, University of Bergen, P.O.Box 7803, N-5020 Bergen, Norway. † lindzey@cs.colostate.edu, Mathematics Department, Colorado State University, Fort Collins, CO, 80523-1873 U.S.A. ‡ rmm@cs.colostate.edu, Computer Science Department, Colorado State University, Fort Collins, CO, 80523-1873 U.S.A. § spin@vuse.vanderbilt.edu, Department of Electrical Engineering and Computer Science, Vander- bilt University, Nashville, TN, U.S.A. ¶ jayme@nce.ufrj.br, Instituto de Matematica, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brasil.