k-Planar and Fan-Crossing Drawings and Transductions of Embeddable Graphs

Abstract

We introduce, for every surface Σ, a two-way connection between FO transductions (first-order logical transformations) of the graphs embeddable in Σ and a certain variant of fan-crossing drawings of graphs in Σ. If the target graphs drawn in Σ are additionally of bounded maximum degree, then the restriction on drawings is simply to have a bounded number of crossings per edge (such as being k-planar for fixed k if Σ is the plane). For graph classes, this connection allows us to derive non-transducibility results from nonexistence of the said drawings and, conversely, from nonexistence of a transduction to derive nonexistence of the said drawings. For example, the class of 3D-grids is not k-planar for any fixed k. We hope that this connection will help to draw a path to a possible proof that not all toroidal graphs are transducible from planar graphs. The result is based on a very recent characterization of weakly sparse FO transductions of classes of bounded expansion by [Gajarský, Gładkowski, Jedelský, Pilipczuk and Toruńczyk, arXiv:2505.15655].

Figures (4)

• Figure 1: (a) An example of a monotone $2$-fold $2$-clustered fan-crossing drawing $D$. The four subdivision vertices (i.e., the set $V(D')\setminus V(D)$ from \ref{['def:clustfan']}) are hollow, and the three components of the crossing graph of the subdivided drawing $D'$ are emphasized by shade colors red, green and blue. (b) An ordinary fan-crossing drawing of a large $m$-vertex graph which is not $1$-fold $\ell$-clustered for any $\ell < \frac{m}{2}$, because no subdivisions are allowed and the crossed edges induce $\lfloor\frac{m}{2}\rfloor$ edge-disjoint fans.
• Figure 2: An illustration of the proof of \ref{['lem:todraw']}. The five blue bags (dashed lines in the picture) show the five model sets of a congestion-$2$ depth-$2$ minor model of a $5$-vertex graph $H$ in the depicted graph $G$ (which is a $3\times5$ square grid pictured in black). The chosen small disks $\delta_u$ and $\delta_e$ at the vertices and edges of $G$ are shaded gray. The set ${\cal R}$ of branching arcs representing the vertices of $H$ (so, one per each blue bag) is drawn with thick red lines, each with its highlighted root, and the shaded disks in which intersections between the arcs of ${\cal R}$ occur are emphasized with darker gray color.
• Figure 3: An illustration of the proof of \ref{['lem:drawingtofo']}(a) (here with $X=\emptyset$); turning a $2$-planar drawing $D$ into a planar colored drawing $D"$. Color $b_0$ is red, and $b_1$ and $b_2$ are green and blue.
• Figure 4: An illustration of the proof of \ref{['lem:drawingtofo']}(b) (here with $X=\emptyset$). (left) Component $M_i$ of the crossing graph of the drawing $D_1$ from the proof, with sets ${\cal F}_i^1$, ${\cal F}_i^2$, ${\cal F}_i^3$ of paths subdividing original fans centered at $w_i^1$, $w_i^2$, $w_i^3$. (right) The corresponding fragment of the embedding $D_3$, displaying the subdivided star $S_i'$ and the coloring assigned by the proof. Color $b_0$ is black, $b_0'$ is white, and $b_j,b_j'$, $j=1,2,3$, are light and dark (resp.) shades of colors in order blue, red and green. The light shades of colors in the picture are given to the vertices of $R_i^j$, and the dark ones to those of $T_i^j$.

Theorems & Definitions (14)

• Theorem 1: Gajarský et al. DBLP:journals/corr/abs-2505.15655
• Definition 2
• Theorem 3
• Corollary 4
• Remark 5
• Lemma 7
• proof
• proof : of '$\Rightarrow$' of \ref{['thm:kfanchar']}
• Lemma 8
• proof