[Paper Review] Counting cocircuits and convex two-colourings is #P-complete

Steven D. Noble, Noble, Steven D.|arXiv (Cornell University)|Oct 11, 2008

Limits and Structures in Graph Theory3 references3 citations

TL;DR

This paper proves that counting convex two-colourings of a graph—where each colour class induces a connected subgraph—and counting cocircuits in a graph are both #P-complete problems. The authors establish a chain of reductions from #Monotone 2-SAT, a known #P-complete problem, through Max Cut, Required Size Cocircuits, Cocircuits, and finally to Convex Two-Colourings, demonstrating computational intractability for both problems via polynomial-time reductions.

ABSTRACT

We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.

Motivation & Objective

• To resolve a question posed by Makowsky regarding the computational complexity of counting convex two-colourings in graphs.
• To establish that counting cocircuits in a graph is #P-complete, a problem not previously formally proven to be so.
• To demonstrate that both problems are computationally intractable by reducing them through a sequence of known #P-complete problems.
• To provide a novel chain of reductions showing the equivalence in complexity between #Monotone 2-SAT, Max Cut, Required Size Cocircuits, Cocircuits, and Convex Two-Colourings.

Proposed method

• Construct a polynomial-time reduction from #Monotone 2-SAT to #Max Cut by building a graph with clause-specific 9-edge circuits connected to a central vertex.
• Transform each #Max Cut instance into a #Required Size Cocircuits instance by adding auxiliary vertices and edges to enforce cocircuit structure and size constraints.
• Use the l-stretch construction of a graph to generate a family of graphs whose cocircuit counts form a system of linear equations with a Vandermonde matrix.
• Apply Gaussian elimination to invert the Vandermonde system and recover the number of cocircuits of each size from the total cocircuit counts of the stretched graphs.
• Establish a direct bijection between cocircuits in a graph and convex two-colourings using both colours, showing that the number of such colourings is twice the number of cocircuits.
• Use the chain of reductions to prove that #Convex Two-Colourings is #P-hard, and since it is in #P, it is #P-complete.

Experimental results

Research questions

• RQ1 Is the problem of counting convex two-colourings of a graph #P-complete?
• RQ2 Is the problem of counting cocircuits in a graph #P-complete?
• RQ3 Can #Max Cut be shown to be #P-complete via a novel reduction from #Monotone 2-SAT?
• RQ4 Can the number of cocircuits of a given size be recovered from the total cocircuit count of graph stretches using linear algebra?
• RQ5 Is there a direct combinatorial correspondence between cocircuits and convex two-colourings in a graph?

Key findings

• The problem of counting convex two-colourings of a graph is #P-complete, resolving an open question by Makowsky.
• The problem of counting cocircuits in a graph is #P-complete, establishing a new #P-completeness result.
• The problem #Max Cut is #P-complete, which is shown via a novel reduction from #Monotone 2-SAT.
• The number of cocircuits of size $k$ in a graph can be recovered in polynomial time from the total cocircuit counts of its $l$-stretches using Gaussian elimination on a Vandermonde matrix.
• For any connected graph, the number of convex two-colourings using both colours is exactly twice the number of cocircuits, providing a direct combinatorial link between the two problems.
• The entire chain of reductions—#Monotone 2-SAT → #Max Cut → #Required Size Cocircuits → #Cocircuits → #Convex Two-Colourings—establishes #P-completeness for all five problems.