Abstract
A graph G = (V,E) is a double-threshold graph if there exist a vertex-weight function w :V rightarrow mathbb {R} and two real numbers mathtt {lb}, mathtt {ub}in mathbb {R} such that uv in E if and only if mathtt {lb}le mathtt {w}(u) + mathtt {w}(v) le mathtt {ub}. In the literature, those graphs are studied also as the pairwise compatibility graphs that have stars as their underlying trees. We give a new characterization of double-threshold graphs that relates them 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 [Algorithmica 2020] ran in O(n^{3} m) time, where n and m are the numbers of vertices and edges, respectively.
Highlights
A graph is a threshold graph if there exist a vertex-weight function and a real number called a weight lower bound such that two vertices are adjacent in the graph if and only if the associated vertex weight sum is at least the weight lower bound
We study the class of double-threshold graphs, which is a proper generalization of threshold graphs and a proper specialization of the graphs that are edge-intersections of two threshold graphs [11]
Our main result in this paper is a linear-time recognition algorithm for double-threshold graphs based on a new characterization
Summary
A graph is a threshold graph if there exist a vertex-weight function and a real number called a weight lower bound such that two vertices are adjacent in the graph if and only if the associated vertex weight sum is at least the weight lower bound. Polynomial-time recognition of double-threshold graphs Xiao and Nagamochi [26] solved the open problem of Calamoneri and Sinaimeri [4] by giving a vertex-ordering characterization and an O(n3m)-time recognition algorithm for star pairwise compatibility graphs, where n an m are the numbers of vertices and edges, respectively. Their result answered the question by Jamison and Sprague [12] about the recognition of double-threshold graphs by the equivalence of the graph classes. We can see that the class of double-threshold graphs connects several other graph classes studied before
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