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arXiv · 2301.01226

$k$-planar Placement and Packing of $\Delta$-regular Caterpillars

Carla Binucci·Emilio Di Giacomo·Michael Kaufmann·Giuseppe Liotta·Alessandra Tappini
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Abstract

This paper studies a \emph{packing} problem in the so-called beyond-planar setting, that is when the host graph is ``almost-planar'' in some sense. Precisely, we consider the case that the host graph is $k$-planar, i.e., it admits an embedding with at most $k$ crossings per edge, and focus on families of $\Delta$-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree $\Delta$. We study the dependency of $k$ from the number $h$ of caterpillars that are packed, both in the case that these caterpillars are all isomorphic to one another (in which case the packing is called \emph{placement}) and when they are not. We give necessary and sufficient conditions for the placement of $h$ $\Delta$-regular caterpillars and sufficient conditions for the packing of a set of $\Delta_1$-, $\Delta_2$-, $\dots$, $\Delta_h$-regular caterpillars such that the degree $\Delta_i$ and the degree $\Delta_j$ of the non-leaf vertices can differ from one caterpillar to another, for $1 \leq i,j \leq h$, $i\neq j$.

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Carla Binucci, Emilio Di Giacomo, Michael Kaufmann, Giuseppe Liotta, Alessandra Tappini. 2023-01-03. $k$-planar Placement and Packing of $\Delta$-regular Caterpillars. https://arxiv.org/abs/2301.01226

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