We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$.

中文翻译：

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计算同态到 $K_4$-minor-free Graphs, modulo 2

我们研究了计算从输入图 $G$ 到固定图 $H$ 的同态数量的奇偶性的问题。Faben 和 Jerrum [ToC'15] 在图 $H$ 上引入了一个明确的标准，并推测，如果满足，则问题可在多项式时间内解决，否则，对于复杂性类 $\oplus\mathrm{ P}$ 的奇偶校验问题。我们验证了他们对所有图 $H$ 的猜想，这些图排除了 $4$ 顶点上的完整图作为次要图。此外，假设随机指数时间假设，我们排除了 $\oplus\mathrm{P}$-complete 情况下亚指数时间算法的存在。我们的证明引入了一种从固定图 $H$ 的全局定义子结构中推导出硬度的新方法。使用这个，