On the Convergence of Some Biclique Operators on Multipartite Graphs

We introduce a new graph operator, called the weak factor graph, which is close to the well-known clique-graph operator but which rather operates in terms of bicliques in a multipartite graph, and we address the problem of the convergence of the series of graphs obtained by the iterative application of this operator. As for the clique-graph operator, it is easy to find graphs for which the series of weak factor graphs does not converge. Here, we show that we can slightly modify the weak-factor-graph operator, into an operator called clean factor graph, so that it converges for all graphs. Moreover, we show that the multipartite graph to which the series converge is a decomposition of a well-known combinatorial object: the inclusion order of the intersections of maximal cliques of the initial graph.