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

TL;DR

This paper investigates the conditions under which multiple $ ext{Delta}$-regular caterpillars can be embedded into $k$-planar graphs, providing necessary and sufficient criteria for placement and packing in a beyond-planar setting.

Contribution

It introduces new necessary and sufficient conditions for placing and packing $ ext{Delta}$-regular caterpillars in $k$-planar graphs, extending beyond traditional planar graph constraints.

Findings

Necessary and sufficient conditions for placement of isomorphic caterpillars.

Sufficient conditions for packing caterpillars with varying degrees.

Analysis of the relationship between the number of caterpillars and the crossing parameter $k$.

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 $1 \leq i,j \leq h$, $i\neq j$.