Linear-Time Recognition of Double-Threshold Graphs
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A graph G = (V,E) is a double-threshold graph if there exist a vertex-weight function w V →R and two real numbers lb, ub∈R such that uv ∈ E if and only if lb<w(u) + w(v) <ub. In the literature, those graphs are studied as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs, which gives connections to bipartite permutation graphs. Using the new characterization, we present a linear-time algorithm for recognizing double-threshold graphs. Prior to our work, the fastest known algorithm by Xiao and Nagamochi [COCOON 2018] ran in O(n^6) time, where n is the number of vertices.
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