Complexity of Anchored Crossing Number and Crossing Number of Almost Planar Graphs

TL;DR

This paper proves that only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as it is shown by [Cabello and Mohar, 2013] to be NP-hard.

Abstract

In this paper we deal with the problem of computing the exact crossing number of almost planar graphs and the closely related problem of computing the exact anchored crossing number of a pair of planar graphs. It was shown by [Cabello and Mohar, 2013] that both problems are NP-hard; although they required an unbounded number of high-degree vertices (in the first problem) or an unbounded number of anchors (in the second problem) to prove their result. Somehow surprisingly, only three vertices of degree greater than 3, or only three anchors, are sufficient to maintain hardness of these problems, as we prove here. The new result also improves the previous result on hardness of joint crossing number on surfaces by [Hliněný and Salazar, 2015]. Our result is best possible in the anchored case since the anchored crossing number of a pair of planar graphs with two anchors each is trivial, and close to being best possible in the almost planar case since the crossing number is efficiently computable for almost planar graphs of maximum degree 3 [Riskin 1996, Cabello and Mohar 2011].

Figures (5)

• Figure 1: a) An example of an anchored graph. Although the graph itself is planar, its anchored crossing number equals $2$. b) Another example made of a disjoint union of two anchored planar graphs. Its anchored crossing number equals $4$.
• Figure 2: An illustration to Corollary \ref{['cor:main3']}; how to translate the problem of the anchored crossing number (of a suitable instance as in \ref{['thm:main3']}) to that of the ordinary crossing number of an almost planar graph. a) The depicted instance consists of a disjoint union of two (blue $P$ and red $R$) anchored planar graphs. b) Turning the previous into an almost planar instance of the ordinary crossing number problem, with a "heavy" cycle on the former anchor vertices, a "flipped out" subdrawing of the component $R$, and an added edge $f$ which effectively forces $P$ and $R$ to stay together inside the cycle in an optimal drawing. c) Blowing up every vertex except the 3 former red anchors into a large cubic grid (a wall), which keeps the drawing properties and stays almost planar.
• Figure 3: A high-level illustration of the assumptions on the PP anchored graph $(H,A)$ in Theorem \ref{['thm:anchorNPHrefin']}; the drawing $\mathcal{D}_1$ of $(H_1,A_1)$ is sketched in red, and the drawing $\mathcal{D}_2$ of $(H_2,A_2)$ is in blue. The blue sketch emphasizes only the "heavy" path $R\subseteq H_2$, while the red sketch shows all paths of the family $\mathcal{Q}$. The solid red paths are all of weight $w$ and the dashed red paths are of minimum weight $w-1$ (but some edges of these paths are also of weight $w$). Note that every vertex of the graph $H_1$ is either a red anchor, or in the intersection of some two paths from $\mathcal{Q}$.
• Figure 6: A schematic picture of the frame PP anchored graph $(F_k,B)$ used for the reduction in \ref{['sec:hardness']} with $k=4$. The solid lines depict edges, while the dashed lines represent paths in general. The thin dashed red paths pairwise intesect in red vertices which are not (and do not need to be) explicitly specified. The red and the blue graphs are vertex-disjoint and each one has three anchors ($r_0,r_2,r_4$ of the red graph, and $b_0,d_1,b_4$ of the blue graph) and is anchored planar. The bracketed numbers represent edge weights, where $[t]$ means weight $\omega^t$. Weights of the edges emphasized with gray shade are treated specially (see a closer detail in \ref{['fig:Fk-order']}). The horizontal red path $P_0$ from $r_0$ to $r_4$ has edges of weight $\omega^{41}+\mathcal{O}(k\omega^{30})$, the horizontal blue path $P_1$ connecting $b_0$ to $b_4$ has, in the subpath from $b_1$ to $b_3$, edges of weight $\omega^{49}+\mathcal{O}(k\omega^{30})$, and elsewhere edges of weight $\omega^{49}+\frac{2}{5}\omega^{35}+\mathcal{O}(k\omega^{30})$; these weights are precisely specified later in the proof. The horizontal blue path $P_2$ connecting $c_0$ to $c_4$ has edges of weight exactly $\omega^{38}$ in the subpath from $c_1$ to $c_3$, and of weight $\omega^{38}+\frac{4}{5}\omega^{35}$ elsewhere. The vertical blue edge $b_2c_2$ is of weight exactly $\omega^{48}+2\omega^{38}-\omega^{34}$, and the vertical red edges $r_1r_1'$ and $r_3r_3'$ are of weight $\omega^{41}-\omega^{40}$.
• Figure 8: An example construction of an almost planar graph $G+uv$ (where the edge $uv$ is in red), such that only the two encircled vertices are of degree higher than $3$, and that the gray regions stand for very dense rigid patches. While $\operatorname{cr}(G+uv)=1$ (just cross the two green edges), the best way of inserting the (red) edge $uv$ into a planar drawing of $G$ can be arbitrarily costly.

Theorems & Definitions (4)

• Theorem 1: Cabello and Mohar DBLP:journals/siamcomp/CabelloM13
• Theorem 2
• Conjecture 3
• Claim 4