K-Planar Placement and Packing of \Delta-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 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 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-, ... , \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.