Open k-monopolies in graphs: complexity and related concepts

Closed monopolies in graphs have a quite long range of applications in several problems related to overcoming failures, since they frequently have some common approaches around the notion of majorities, for instance to consensus problems, diagnosis problems or voting systems. We introduce here open k-monopolies in graphs which are closely related to different parameters in graphs. Given a graph G = (V,E) and X ⊆ V , if δX(v) is the number of neighbors v has in X, k is an integer and t is a positive integer, then we establish in this article a connection between the following three concepts: • Given a nonempty set M ⊆ V a vertex v of G is said to be k-controlled by M if δM (v) ≥ δV (v) 2 + k. The set M is called an open k-monopoly for G if it k-controls every vertex v of G. ∗Supported by the Ministry of Science of Slovenia under the grant P1-0297. The author is also with IMFM, Jadranska 19, 1000 Ljubljana, Slovenia. †The research was partially done while the author was at University of Maribor, Slovenia, supported by “Ministerio de Educación, Cultura y Deporte”, Spain, under the “Jose Castillejo” program for young researchers. Reference number: CAS12/00267.