The complexity of counting planar graph homomorphisms of domain size 3

TL;DR

This paper establishes a complete complexity classification for counting planar graph homomorphisms with domain size 3, identifying which cases are polynomial-time solvable and which are P-hard, using analytic methods.

Contribution

It provides a full dichotomy theorem for the complexity of counting planar graph homomorphisms of size 3, extending understanding to general domain sizes and employing analytic techniques.

Findings

Problems are either polynomial-time solvable or P-hard.

Characterization includes cases solvable by holographic algorithms.

Extends results to general domain sizes with analytic proofs.

Abstract

We prove a complexity dichotomy theorem for counting planar graph homomorphisms of domain size 3. Given any 3 by 3 real valued symmetric matrix $H$ defining a graph homomorphism from all planar graphs $G \mapsto Z_H(G)$, we completely classify the computational complexity of this problem according to the matrix $H$. We show that for every $H$, the problem is either polynomial time computable or \#P-hard. The P-time computable cases consist of precisely those that are P-time computable for general graphs (a complete classification is known) or computable by Valiant's holographic algorithm via matchgates. We also prove several results about planar graph homomorphisms for general domain size $q$. The proof uses mainly analytic arguments.