Simple Realizability of Abstract Topological Graphs

Abstract

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 $Γ_A$ of $G$ in the plane such that any two edges $e_1,e_2$ of $G$ cross in $Γ_A$ if and only if $(e_1,e_2) \in \mathcal{X}$; $Γ_A$ 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 $\mathrmλ(A)$ 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 $\mathrmλ(A) \geq 6$. On the positive side, we give an optimal linear-time algorithm that solves SATR when $\mathrmλ(A) \leq 3$ 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.

Figures (17)

• Figure 1: Illustrations for the proof of \ref{['lem:untangling']}. (a) A schematic representation of a simple realization $\Gamma_A$ of an AT-graph $A$ with a tangled $3$-crossing $E' = \{e_1, e_2, e_3\}$. (b) The simple realization $\Gamma'_A$ obtained from $\Gamma_A$, where $E'$ is untangled. (c) The curves forming $e_1$ in $\Gamma'_A$.
• Figure 2: PQ-trees representing the circular orders $(a)$$(1, 2, 3, 4)$ and its reverse, $(b)$ of $1, 2, 3, 4, 5$, such that $1, 2$ are consecutive, $(c)$$(6, 1, 2, 3, 4, 5)$, $(6, 1, 5, 3, 4, 2)$, $(1, 6, 2, 4, 3, 5)$, $(1, 6, 5, 4, 3, 2)$.
• Figure 3: A graph $H$ and its SPQR-tree decomposition (virtual edges drawn in gray), where Q-nodes are omitted in favor of allowing real edges (drawn in black) in the skeletons of the other nodes.
• Figure 5: (a,b) The split gadget $S$ : The clockwise circular order of the edges leaving the gadget is either $b_1$, $f_1$, $b_2$, $f_2$, $b_3$, $f_3$ (a) or $f_1$, $b_1$, $f_2$, $b_2$, $f_3$, $b_3$ (b). (c) The variable gadget $\mathcal{V}_v$. The dashed edges belong to the variable cycle of $v$ in the skeleton $H_\varphi$.
• Figure 6: Illustration of the non-trivial connected components of (a) the crossing graph of the split gadget $S$ and (b) the crossing graph of the clause gadget $\mathcal{Q}_c$. (c) The non-trivial connected components of $\mathcal{C}(A_\varphi)$ involving an edge of the variable cycle (left) and of the clause cycle (right).

Theorems & Definitions (21)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 7
• Corollary 8
• Lemma 10
• Lemma 11