k-Planar Placement and Packing of Δ-Regular Caterpillars

This paper studies a 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 [Formula: see text]-planar, i.e., it admits an embedding with at most [Formula: see text] crossings per edge, and focus on families of [Formula: see text]-regular caterpillars, that are caterpillars whose non-leaf vertices have the same degree [Formula: see text]. We study the dependency of [Formula: see text] from the number [Formula: see text] 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 placement) and when they are not. We give necessary and sufficient conditions for the placement of [Formula: see text] [Formula: see text]-regular caterpillars and sufficient conditions for the packing of a set of [Formula: see text]-, [Formula: see text]-, [Formula: see text], [Formula: see text]-regular caterpillars such that the degree [Formula: see text] and the degree [Formula: see text] of the non-leaf vertices can differ from one caterpillar to another, for [Formula: see text], [Formula: see text].