Dynamically Operating on Threshold Graphs and Related Classes (Extended Abstract)

A graph G = (V,E) is a threshold graph if there is a mapping a : V → R+ and a positive real number S such that a(v) < S for all v ∈ V and (v, w) ∈ E if and only if a(v) + a(w) ≥ S. It is well known that G can be partitioned into a clique K and a stable set I. Threshold graphs constitute a very important and well studied class in graph theory and graph algorithms, since they have applications in several fields, such as psychology, parallel processing, scheduling. For this reason, threshold graphs have been defined many times in the literature, with different names (see, e.g. [4,6,7]). A similar definition describes the class of difference graphs (also known as chain graphs): a graph G = (V,E) is a difference graph if there is mapping a : V → R and a positive real number T such that |a(v)| < T for all v ∈ V and (v, w) ∈ E if and only if |a(v)− a(w)| ≥ T . Difference and threshold graphs are incomparable; difference graphs have been also independently introduced several times, for example in [5,8,12]. For a comprehensive survey on threshold graphs, difference graphs and related topics, see [9].