Note on Min-k-Planar Drawings of Graphs

TL;DR

The paper investigates min-$k$-planar drawings, where crossing pairs must include at least one edge with at most $k$ crossings, and contrasts general versus simple drawings. It proves that for arbitrarily large fixed $k$, there exist graphs with a min-$2$-planar drawing (or min-$3$-planar drawing avoiding adjacent-edge crossings) that admit no simple min-$k$-planar drawing, using a frame-based construction built from $t$-amplifications and anchored graphs. The core technique enforces prescribed anchored subdrawings and yields explicit counterexamples via anchored graphs $(G_k,A_k)$ and $(G'_k,A'_k)$. This reveals a sharp divergence between simple and general min-$k$-planar drawings (beyond the trivial $k=1$ case) and motivates a careful examination of which prior results extend to the general setting, with the frame tool proposed for future exploration.

Abstract

The k-planar graphs, which are (usually with small values of k such as 1, 2, 3) subject to recent intense research, admit a drawing in which edges are allowed to cross, but each one edge is allowed to carry at most k crossings. In recently introduced [Binucci et al., GD 2023] min-k-planar drawings of graphs, edges may possibly carry more than k crossings, but in any two crossing edges, at least one of the two must have at most k crossings. In both concepts, one may consider general drawings or a popular restricted concept of drawings called simple. In a simple drawing, every two edges are allowed to cross at most once, and any two edges which share a vertex are forbidden to cross. While, regarding the former concept, it is for k<=3 known (but perhaps not widely known) that every general k-planar graph admits a simple k-planar drawing and this ceases to be true for any k>=4, the difference between general and simple drawings in the latter concept is more striking. We prove that there exist graphs with a min-2-planar drawing, or with a min-3-planar drawing avoiding crossings of adjacent edges, which have no simple min-k-planar drawings for arbitrarily large fixed k.

Figures (3)

• Figure 1: An illustration of \ref{['prop:tosimple']} a); up to symmetry between $e$ and $f$ (with common end vertex $x$), the edge $e$ carries no other crossing than the point $y$, and so one can draw an uncrossed arc $e'_1$ tightly along the segment $e_1\subseteq e$ from $x$ to $y$. When redrawing from $e$ to $e"$ and from $f$ to $f"$ (using $e'_1$), the crossing at $y$ is eliminated and no new crossings are added to any edge in the picture.
• Figure 2: An illustration of \ref{['prop:tosimple']} b); now we have two crossings $x$ and $y$ of the same pair $e$ and $f$ of edges, and there are no more crossings on $e$. Similarly to \ref{['fig:simplif1']}, when redrawing from $e$ to $e"$ and from $f$ to $f"$ (using the uncrossed arc $e'_1$), at least one of the crossings at $x$ or $y$ is eliminated and no new crossings are added to any edge in the picture.
• Figure 5: An illustration of the graph $H_2$ from the proof of \ref{['lem:fullfram']}. The three black circled vertices are the designated anchors $A$ (here $A=\{w_2^1,w_2^2,w_2^3\}$). The subgraph $H_2'$ is in thick black and $H_2"$ in magenta colour. All magenta edges get $t$-amplified in the construction of $H$, and consequently, the black edges forced to cross them will be heavy in any min-$k$-planar drawing of $H$ by \ref{['lem:amplify']}.

Theorems & Definitions (5)

• Theorem 1: Proof in \ref{['sec:details']}
• Proposition 2
• Lemma 3
• Lemma 4
• Lemma 5