An abstract topological graph (AT-graph) is a pair $A=(G,\mathcal{X})$, where $G=(V,E)$ is a graph and $\mathcal{X} \subseteq {E \choose 2}$ is a set of pairs of edges of $G$. A realization of $A$ is a drawing $\Gamma_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $\Gamma_A$ if and only if $(e_1,e_2) \in \mathcal{X}$; $\Gamma_A$ is simple if any two edges intersect at most once (either at a common endpoint or at a proper crossing). The AT-graph Realizability (ATR) problem asks whether an input AT-graph admits a realization. The version of this problem that requires a simple realization is called Simple AT-graph Realizability (SATR). It is a classical result that both ATR and SATR are NP-complete. In this paper, we study the SATR problem from a new structural perspective. More precisely, we consider the size $\mathrm{\lambda}(A)$ of the largest connected component of the crossing graph of any realization of $A$, i.e., the graph ${\cal C}(A) = (E, \mathcal{X})$. This parameter represents a natural way to measure the level of interplay among edge crossings. First, we prove that SATR is NP-complete when $\mathrm{\lambda}(A) \geq 6$. On the positive side, we give an optimal linear-time algorithm that solves SATR when $\mathrm{\lambda}(A) \leq 3$ and returns a simple realization if one exists. Our algorithm is based on several ingredients, in particular the reduction to a new embedding problem subject to constraints that require certain pairs of edges to alternate (in the rotation system), and a sequence of transformations that exploit the interplay between alternation constraints and the SPQR-tree and PQ-tree data structures to eventually arrive at a simpler embedding problem that can be solved with standard techniques.

翻译：抽象拓扑图（AT-图）是一个二元组 $A=(G,\mathcal{X})$，其中 $G=(V,E)$ 是一个图，而 $\mathcal{X} \subseteq {E \choose 2}$ 是 $G$ 的边对集合。$A$ 的一个实现是 $G$ 在平面上的一个绘制 $\Gamma_A$，使得 $G$ 的任意两条边 $e_1,e_2$ 在 $\Gamma_A$ 中相交当且仅当 $(e_1,e_2) \in \mathcal{X}$；如果任意两条边至多相交一次（要么在公共端点，要么在正常交叉点），则称 $\Gamma_A$ 是简单的。AT-图可实现性（ATR）问题询问一个给定的 AT-图是否允许一个实现。要求实现是简单的该问题版本称为简单 AT-图可实现性（SATR）。经典结论表明 ATR 和 SATR 都是 NP 完全问题。在本文中，我们从一种新的结构视角研究 SATR 问题。具体来说，我们考虑 $A$ 的任何实现的交叉图的最大连通分量的大小 $\mathrm{\lambda}(A)$，即图 ${\cal C}(A) = (E, \mathcal{X})$。该参数代表了一种衡量边交叉之间相互作用程度的自然方式。首先，我们证明当 $\mathrm{\lambda}(A) \geq 6$ 时 SATR 是 NP 完全的。从积极方面看，我们给出了一个最优的线性时间算法，当 $\mathrm{\lambda}(A) \leq 3$ 时该算法可解决 SATR 问题，并在存在时返回一个简单实现。我们的算法基于若干要素，特别是归约到一个新的嵌入问题，该问题受限于要求特定边对在旋转系统中交替出现的约束，以及一系列利用交替约束与 SPQR 树和 PQ 树数据结构之间相互作用进行的变换，最终得到一个可用标准技术解决的更简单的嵌入问题。