The Complexity of $(P_k, P_\ell)$-Arrowing

TL;DR

This paper investigates the computational complexity of the $(P_k, P_ll)$-Arrowing problem, establishing coNP-completeness for most pairs of $k$ and $ll$, and providing a polynomial-time solution for the specific case $(P_3, P_4)$.

Contribution

It proves coNP-completeness for nearly all pairs of $k$ and $ll$, introduces the concept of transmitters for reductions, and offers a polynomial-time algorithm for $(P_3, P_4)$-Arrowing.

Findings

Most $(P_k, P_ll)$-Arrowing problems are coNP-complete.

The problem is polynomial-time solvable for $(P_3, P_4)$.

Introduces transmitters as a new tool for complexity reductions.

Abstract

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.