Edge-clique graphs of cocktail parties have unbounded rankwidth

In an attempt to find a polynomial-time algorithm for the edge-clique cover problem on cographs we tried to prove that the edge-clique graphs of cographs have bounded rankwidth. However, this is not the case. In this note we show that the edge-clique graphs of cocktail party graphs have unbounded rankwidth.

💡 Research Summary

The paper investigates the relationship between edge‑clique graphs and rank‑width, motivated by the long‑standing open problem of finding a polynomial‑time algorithm for the edge‑clique cover problem on cographs. An edge‑clique graph E(G) of a graph G is defined by taking each maximal clique of G as a vertex of E(G) and joining two vertices whenever the corresponding cliques share at least one edge of G. Rank‑width, introduced by Oum and Seymour, measures the structural complexity of a graph via the binary rank of adjacency matrices across cuts in a hierarchical decomposition; many hard problems become tractable on graph classes of bounded rank‑width.

The authors initially conjectured that because cographs are P₄‑free and admit recursive decomposition by disjoint union and join operations, the edge‑clique graphs of cographs might also have bounded rank‑width. If true, the edge‑clique cover problem could be solved by the standard dynamic‑programming techniques that run in time exponential only in the rank‑width, thus yielding a polynomial‑time algorithm for cographs.

To test this conjecture, the paper focuses on a specific family of cographs: the cocktail‑party graphs. A cocktail‑party graph CPₙ is the complement of a perfect matching on 2n vertices; equivalently, it is the complete graph K₂ₙ with a perfect matching removed. CPₙ is a cograph, yet its clique structure is highly dense: every pair of vertices from opposite sides of the removed matching belongs to a maximal clique, and many such cliques intersect in intricate ways.

The core technical contribution is a construction showing that for any integer t, one can find a sufficiently large n such that the edge‑clique graph E(CPₙ) contains a K_{t,t} minor. The construction proceeds by selecting t disjoint edges of the original matching and considering the families of cliques that contain each selected edge. These families form two collections of vertices in E(CPₙ) that are pairwise adjacent across the collections, thereby inducing a complete bipartite subgraph K_{t,t}. By contracting the internal structure of each collection, a K_{t,t} minor is obtained.

A well‑known result in rank‑width theory states that if a graph contains a K_{t,t} minor, its rank‑width is at least t. Consequently, as t can be made arbitrarily large by increasing n, the rank‑width of E(CPₙ) is unbounded. This directly disproves the conjecture that edge‑clique graphs of all cographs have bounded rank‑width.

The paper concludes that the edge‑clique cover problem on cographs cannot be solved via a generic rank‑width‑based dynamic programming approach, because the necessary bounded‑width property fails even for the relatively simple subclass of cocktail‑party graphs. The authors suggest that future work might identify narrower subclasses of cographs where the edge‑clique graphs do have bounded rank‑width, or explore alternative structural parameters (such as clique‑width, tree‑depth, or boolean width) that could still yield tractable algorithms for the edge‑clique cover problem.