The Complexity of (P_k, P_ℓ)-Arrowing
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For fixed nonnegative integers k and ℓ, the (P_k, P_ℓ)-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_ℓ. The problem is trivial when max(k,ℓ) ≤ 3, but has been shown to be coNP-complete when k = ℓ = 4. In this work, we show that the problem remains coNP-complete for all pairs of k and ℓ, except (3,4), and when max(k,ℓ) ≤ 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 ℓ≥ 5, 2) ℓ > k ≥ 4, and 3) ℓ = k ≥ 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_ℓ)-Arrowing problems.
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