Planar Graph Homomorphisms: A Dichotomy and a Barrier from Quantum Groups

TL;DR

This work studies the complexity of counting planar graph homomorphisms parameterized by a symmetric nonnegative matrix $M$, introducing a rich gadget framework that preserves planarity. It proves a sharp dichotomy for matrices with pairwise distinct diagonals, placing $Pl\text{-}GH(M)$ either in $P$ or $\#\mathsf{P}$-hard via a simple, checkable criterion and extends dichotomy results to cases where planar edge gadgets separate vertex labels. A central bridge is made between planar gadget expressivity and quantum automorphism groups $\mathrm{Qut}(M)$, showing planar separability occurs exactly when $\mathrm{Qut}(M)$ is trivial, and proving this triviality problem is undecidable. Collectively, the results delineate a barrier to transferring nonplanar reduction techniques to the planar setting and chart concrete directions for future work on diagonal/non-diagonal and quantum-symmetry regimes.

Abstract

We study the complexity of counting (weighted) planar graph homomorphism problem $\tt{Pl\text{-}GH}(M)$ parametrized by an arbitrary symmetric non-negative real valued matrix $M$. For matrices with pairwise distinct diagonal values, we prove a complete dichotomy theorem: $\tt{Pl\text{-}GH}(M)$ is either polynomial-time tractable, or $\#$P-hard, according to a simple criterion. More generally, we obtain a dichotomy whenever every vertex pair of the graph represented by $M$ can be separated using some planar edge gadget. A key question in proving complexity dichotomies in the planar setting is the expressive power of planar edge gadgets. We build on the framework of Mančinska and Roberson to establish links between \textit{planar} edge gadgets and the theory of the \textit{quantum automorphism group} $\tt{Qut}(M)$. We show that planar edge gadgets that can separate vertex pairs of $M$ exist precisely when $\tt{Qut}(M)$ is \emph{trivial}, and prove that the problem of whether $\tt{Qut}(M)$ is trivial is undecidable. These results delineate the frontier for planar homomorphism counting problems and uncover intrinsic barriers to extending nonplanar reduction techniques to the planar setting.

Figures (5)

• Figure 1: A graph $G$, the thickening gadget $\mathbf{T_{4}}$, and $\mathbf{T_{4}}G$.
• Figure 2: A graph $G$, the stretching gadget $\mathbf{S_{4}}$, and $\mathbf{S_{4}}G$.
• Figure 3: A graph $G$, the bridge gadget $\mathbf{B_{4}}$, and $\mathbf{B_{4}}G$.
• Figure 4: A graph $G$, the loop gadget $\mathbf{L_{3}}$, and $\mathbf{L_{3}}G$.
• Figure 5: A graph $G$, the ring gadget $\mathbf{R_{2, 3}}$ on vertices of degree $3$ and $2$, and $\mathbf{R_{2, 3}}G$.

Theorems & Definitions (150)

• Definition 1: $\mathop{\mathrm{\mathfrak{E}}}\nolimits(M)$, $\mathop{\mathrm{\tt{Pl}\text{-}\mathfrak{E}}}\nolimits(M)$
• Definition 2
• Lemma 2
• Definition 2
• Remark
• Definition 3
• Lemma 3
• Lemma 3
• Definition 4
• Lemma 4