[Paper Review] Edge-clique graphs of cocktail parties have unbounded rankwidth

Maw‐Shang Chang, Ton Kloks+1 more|arXiv (Cornell University)|May 11, 2012

Advanced Graph Theory Research27 references4 citations

TL;DR

This paper demonstrates that edge-clique graphs of cocktail party graphs have unbounded rankwidth, disproving the conjecture that edge-clique graphs of cographs have bounded rankwidth. The authors use a contradiction argument based on χ-boundedness and the independence number of edge-clique graphs, showing that if rankwidth were bounded, it would contradict known lower bounds on the edge-clique covering number and independence number in these graphs.

ABSTRACT

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.

Motivation & Objective

• To investigate whether edge-clique graphs of cographs have bounded rankwidth, a key structural property for efficient algorithms.
• To determine the rankwidth behavior of edge-clique graphs derived from cocktail party graphs, a specific class of cographs.
• To challenge the feasibility of polynomial-time algorithms for the edge-clique cover problem on cographs via structural graph theory.
• To explore the implications of unbounded rankwidth for parameterized complexity and kernelization of the edge-clique cover problem.

Proposed method

• Define the edge-clique graph $K_e(G)$ as having vertices corresponding to edges of $G$, with adjacency when edges share a common clique in $G$.
• Use the fact that $\theta_e(G) = \kappa(K_e(G))$, where $\kappa$ is the vertex-clique cover number.
• Apply Theorem 1.1 to establish a logarithmic lower bound on $\theta_e(cp(n))$, implying $\theta_e(cp(n)) \geq \log_2(2n+1)$.
• Establish that $\alpha(K_e(cp(n))) = 4$ for $n \geq 2$, using explicit construction and adjacency analysis in the edge-clique graph.
• Leverage the result that rankwidth of a graph is at most one more than that of its complement to relate $\text{rankwidth}(K_e(G))$ to $\text{rankwidth}(\overline{K_e(G)})$.
• Use Dvořák and Král’s result that graphs of bounded rankwidth are χ-bounded to derive a contradiction if $K_e(cp(n))$ had bounded rankwidth.

Experimental results

Research questions

• RQ1 Do edge-clique graphs of cographs have bounded rankwidth?
• RQ2 What is the rankwidth of edge-clique graphs derived from cocktail party graphs?
• RQ3 Can the edge-clique cover problem on cographs be solved in polynomial time if edge-clique graphs have bounded rankwidth?
• RQ4 Is there a contradiction between the χ-boundedness of bounded-rankwidth graphs and the independence number and clique cover number of $K_e(cp(n))$?
• RQ5 Does the unbounded rankwidth of $K_e(cp(n))$ imply limitations on kernelization for the edge-clique cover problem?

Key findings

• The edge-clique graphs of cocktail party graphs have unbounded rankwidth, disproving the hypothesis that edge-clique graphs of cographs have bounded rankwidth.
• For $n \geq 2$, the independence number $\alpha(K_e(cp(n)))$ is exactly 4, a key structural invariant.
• The edge-clique covering number $\theta_e(cp(n))$ satisfies $\theta_e(cp(n)) \geq \log_2(2n+1)$, growing logarithmically with $n$.
• The contradiction arises because if $K_e(cp(n))$ had bounded rankwidth, then $\kappa(K_e(cp(n)))$ would be bounded by a function of $\alpha(K_e(cp(n))) = 4$, which contradicts the logarithmic growth of $\theta_e(cp(n))$.
• The result implies that the class of edge-clique graphs of cographs is not of bounded rankwidth, limiting the applicability of rankwidth-based algorithms.
• The edge-clique cover problem remains NP-complete for cographs, as suggested by the structural complexity of their edge-clique graphs.