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.

翻译：我们研究了由任意对称非负实值矩阵$M$参数化的（加权）平面图同态计数问题$\tt{Pl\text{-}GH}(M)$的复杂度。对于对角线条目两两不同的矩阵，我们证明了一个完整的二分定理：根据一个简单的判据，$\tt{Pl\text{-}GH}(M)$要么是多项式时间可解的，要么是$\#$P-困难的。更一般地，只要矩阵$M$所表示的图的任意顶点对都能被某个平面边构件所分离，我们就能得到一个二分定理。证明平面情形下复杂度二分定理的一个关键问题是平面边构件的表达能力。我们基于Mančinska和Roberson的框架，建立了\textit{平面}边构件与\textit{量子自同构群}$\tt{Qut}(M)$理论之间的联系。我们证明了能够分离$M$的顶点对的平面边构件存在，当且仅当$\tt{Qut}(M)$是\emph{平凡的}，并证明了判断$\tt{Qut}(M)$是否平凡的问题是不可判定的。这些结果划定了平面同态计数问题的前沿，并揭示了将非平面归约技术扩展到平面情形所固有的障碍。