Abstract
AbstractThe threshold weight of a graph G is introduced as a measure of the amount by which G differs from being a threshold graph. The threshold graphs are precisely the graphs whose threshold weights are 0. At the opposite extreme is the class of graphs for which the threshold weight is the maximum possible. Such graphs are defined as heavy graphs. Among the results are as following: A theorem that specifies the threshold weight of any triangle‐free graph; necessary and sufficient conditions for a heavy graph in terms of the solvability of a system of linear inequalities; some sufficient conditions for a graph to be heavy and a necessary condition (conjectured to be sufficient, as well) for a heavy graph in terms of its cliques.
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