Abstract
A graph \(G = (V,E)\) is a threshold tolerance graph if each vertex \(v \in V\) can be assigned a weight \(w_v\) and a tolerance \(t_v\) such that two vertices \(x,y \in V\) are adjacent if \(w_x + w_y \ge \min (t_x,t_y)\). 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 their complements, the co-threshold tolerance (co-TT) graphs, resolving an open question of Golumbic, Weingarten, and Limouzy.
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Golovach, P.A., Heggernes, P., Lindzey, N., McConnell, R.M., dos Santos, V.F., Spinrad, J.P. (2014). Recognizing Threshold Tolerance Graphs in \(O(n^2)\) Time. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_18
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