[Paper Review] Cliques in Odd-Minor-Free Graphs

Ken‐ichi Kawarabayashi, David R. Wood|arXiv (Cornell University)|Aug 15, 2011

Advanced Graph Theory Research33 references3 citations

TL;DR

This paper establishes that odd-minor-free graphs contain at most $O(n^2)$ cliques, a bound that is tight due to the complete bipartite graph $K_{n,n}$. It presents an $O(n^3)$-time algorithm to list all cliques in such graphs, leveraging structural decomposition and a recursive clique enumeration method.

ABSTRACT

This paper is about: (1) bounds on the number of cliques in a graph in a particular class, and (2) algorithms for listing all cliques in a graph. We present a simple algorithm that lists all cliques in an $n$-vertex graph in O(n) time per clique. For O(1)-degenerate graphs, such as graphs excluding a fixed minor, we describe a O(n) time algorithm for listing all cliques. We prove that graphs excluding a fixed odd-minor have $O(n^2)$ cliques (which is tight), and conclude a $O(n^3)$ time algorithm for listing all cliques.

Motivation & Objective

• To establish tight upper bounds on the number of cliques in odd-minor-free graphs.
• To develop an efficient algorithm for listing all cliques in odd-minor-free graphs.
• To extend known results on clique enumeration from minor-free graphs to the broader class of odd-minor-free graphs.
• To demonstrate that the $O(n^2)$ clique bound is tight by exhibiting extremal examples such as $K_{n,n}$.

Proposed method

• Uses a recursive clique enumeration algorithm that, for each vertex, lists all cliques containing it and all cliques not containing it.
• Applies structural decomposition theorems to represent odd-minor-free graphs as $(\leq h)$-sums of $h$-almost bipartite or $h$-almost surface-embeddable graphs.
• Employs induction on graph decomposition trees to bound the number of cliques in each component and their combinations.
• Applies a key lemma showing that if each component has $O(n_i^2)$ cliques and no $k$-clique, then the combined graph has $O(n^2)$ cliques.
• Uses the fact that odd-minor-free graphs exclude a fixed odd minor, enabling bounded expansion and controlled clique growth.
• Combines the clique enumeration algorithm with the $O(n^2)$ bound to achieve an $O(n^3)$ total time complexity for listing all cliques.

Experimental results

Research questions

• RQ1 What is the maximum number of cliques in an $n$-vertex odd-minor-free graph?
• RQ2 Can an efficient algorithm list all cliques in odd-minor-free graphs, and what is its time complexity?
• RQ3 Is the $O(n^2)$ bound on the number of cliques in odd-minor-free graphs tight?
• RQ4 How does the clique structure in odd-minor-free graphs compare to that in minor-free graphs?
• RQ5 Can structural decomposition theorems be leveraged to bound clique counts and enable efficient enumeration?

Key findings

• Every $n$-vertex odd-$H$-minor-free graph contains at most $cn^2$ cliques for some constant $c$ depending on $H$, and this bound is tight.
• $K_{n,n}$, which is odd-$K_3$-minor-free, contains $\Theta(n^2)$ cliques, proving the $O(n^2)$ bound is asymptotically tight.
• An $O(n^3)$-time algorithm exists to list all cliques in any $n$-vertex odd-minor-free graph.
• The number of cliques in such graphs is polynomially bounded, in contrast to general graphs which can have $2^n$ cliques.
• The structural decomposition of odd-minor-free graphs into $h$-almost bipartite or $h$-almost surface-embeddable components enables clique counting via induction.
• The recursive clique listing algorithm runs in $O(n)$ time per clique, and when combined with the $O(n^2)$ clique bound, yields an $O(n^3)$ total runtime.