Linear-Time Recognition of Double-Threshold Graphs

Abstract

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.

Characterization and Linear-time Recognition of Paired Threshold Graphs

“…They admit several nice characterizations, including inductive construction, degree sequences, forbidden induced subgraphs (Figure 1), to name a few [13]. Relaxing these characterizations in one way or another, we end with several graph classes, e.g., cographs, split graphs, trivially perfect graphs and double-threshold graphs [14,4,6,11]. Yet another closely related graph class is the difference graphs, defined solely by weight differences [8].…”