Linear Separation of Dominating Sets in Graphs

The class of finite undirected graphs G having the property that there exist real positive numbers associated to their vertices so that a set of vertices is dominating if and only if the sum of the corresponding weights exceeds a certain threshold θ is characterized: (a) by forbidden induced subgraphs; (b) by the linearity of a certain partial order on the vertices of G; (c) by the global structure of G. The class properly includes that of threshold graphs and is properly included in that of perfect graphs.