Packing Trees into 1-planar Graphs - De Luca, Felice; Di Giacomo, Emilio; Hong, Seok-Hee; Kobourov, Stephen; Lenhart, William; Liotta, Giuseppe; Meijer, Henk; Tappini, Alessandra; Wismath, Stephen (Journal of Graph Algorithms and Applications, 2021)
We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, …, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with $n ≥ 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n ≤ 10$.