On the Termination of Some Biclique Operators on Multipartite Graphs

TL;DR

This paper introduces the weak-factor graph operator for multipartite graphs, investigates its termination properties, and proposes a modified operator, the clean-factor, which guarantees termination for all graphs, revealing a rich combinatorial structure.

Contribution

It defines the weak-factor operator, analyzes its non-termination cases, and introduces the clean-factor operator ensuring termination with a detailed combinatorial characterization.

Findings

The weak-factor operator may not terminate on some graphs.

The clean-factor operator always terminates for any input graph.

The terminating graphs have a structure linked to chains of clique intersections.

Abstract

We 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.