$k$-planar Placement and Packing of $Δ$-regular Caterpillars

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 $Δ$-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree $Δ$. 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 $h$ $Δ$-regular caterpillars and sufficient conditions for the packing of a set of $Δ_1$-, $Δ_2$-, $\dots$, $Δ_h$-regular caterpillars such that the degree $Δ_i$ and the degree $Δ_j$ 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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