Journal of Graph Algorithms and Applications

Special Issue on Selected Papers from the 14th International Conference and Workshops on Algorithms and Computation, WALCOM 2020

DOI: 10.7155/jgaa.00574

Packing Trees into 1-planar Graphs

Felice De Luca, Emilio Di Giacomo, Seok-Hee Hong, Stephen Kobourov, William Lenhart, Giuseppe Liotta, Henk Meijer, Alessandra Tappini, and Stephen Wismath

Vol. 25, no. 2, pp. 605-624, 2021. Regular paper.

Abstract We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, 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 $n \geq 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n \leq 10$.