For fixed nonnegative integers $k$ and $\ell$, the $(P_k, P_\ell)$-Arrowing problem asks whether a given graph, $G$, has a red/blue coloring of $E(G)$ such that there are no red copies of $P_k$ and no blue copies of $P_\ell$. The problem is trivial when $\max(k,\ell) \leq 3$, but has been shown to be coNP-complete when $k = \ell = 4$. In this work, we show that the problem remains coNP-complete for all pairs of $k$ and $\ell$, except $(3,4)$, and when $\max(k,\ell) \leq 3$. Our result is only the second hardness result for $(F,H)$-Arrowing for an infinite family of graphs and the first for 1-connected graphs. Previous hardness results for $(F, H)$-Arrowing depended on constructing graphs that avoided the creation of too many copies of $F$ and $H$, allowing easier analysis of the reduction. This is clearly unavoidable with paths and thus requires a more careful approach. We define and prove the existence of special graphs that we refer to as ``transmitters.'' Using transmitters, we construct gadgets for three distinct cases: 1) $k = 3$ and $\ell \geq 5$, 2) $\ell > k \geq 4$, and 3) $\ell = k \geq 4$. For $(P_3, P_4)$-Arrowing we show a polynomial-time algorithm by reducing the problem to 2SAT, thus successfully categorizing the complexity of all $(P_k, P_\ell)$-Arrowing problems.

翻译：对于固定的非负整数 $k$ 和 $\ell$，$(P_k, P_\ell)$-箭形问题询问：给定图 $G$，是否存在对 $E(G)$ 的红/蓝着色，使得图中既无红色 $P_k$ 也无蓝色 $P_\ell$。当 $\max(k,\ell) \leq 3$ 时该问题平凡，但已被证明当 $k = \ell = 4$ 时为 coNP-完全。本文证明，除 $(3,4)$ 以及 $\max(k,\ell) \leq 3$ 的情形外，该问题对任意 $k$ 和 $\ell$ 的组合均为 coNP-完全。这一结果是 $(F,H)$-箭形问题领域针对无限图族的第二个困难性结论，也是首个关于 1-连通图的结果。此前 $(F,H)$-箭形问题的困难性结论依赖于构造避免产生过多 $F$ 和 $H$ 副本的图，从而简化归约分析。但这一方法对于路径结构显然不可行，因此需要更精巧的处理。我们定义并证明了称为“发射器”的特殊图的存在性。利用发射器，我们为三种不同情形构造了子图：1) $k = 3$ 且 $\ell \geq 5$；2) $\ell > k \geq 4$；3) $\ell = k \geq 4$。对于 $(P_3, P_4)$-箭形问题，我们通过将其归约为 2SAT 给出了多项式时间算法，从而完整刻画了所有 $(P_k, P_\ell)$-箭形问题的计算复杂性。