On the termination of some biclique operators on multipartite graphs

Abstract

International audienceWe define a new graph operator, called the weak-factor graph, which comes from the context of complex network modelling. The weak-factor operator is close to the well-known clique-graph operator but it rather operates in terms of bicliques in a multipartite graph. We address the problem of the termination of the series of graphs obtained by iteratively applying the weak-factor operator starting from a given input graph. As for the clique-graph operator, it turns out that some graphs give rise to series that do not terminate. Therefore, we design a slight variation of the weak-factor operator, called clean-factor, and prove that its associated series terminates for all input graphs. In addition, we show that the multipartite graph on which the series terminates has a very nice combinatorial structure: we exhibit a bijection between its vertices and the chains of the inclusion order on the intersections of the maximal cliques of the input graph

On closure operators related to maximal tricliques in tripartite hypergraphs

“…There is an interesting connection between biclique operators, their associated graphs (Crespelle et al, 2015), and switching generators; in these graphs, two vertices (maximal biclques) are connected if they have a non-empty intersection, which, under some conditions, can be the switching generator of those biclques, i.e. concepts.…”