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Published in:
2020 | OriginalPaper | Chapter
Packing Trees into 1-Planar Graphs
Authors : Felice De Luca, Emilio Di Giacomo, Seok-Hee Hong, Stephen Kobourov, William Lenhart, Giuseppe Liotta, Henk Meijer, Alessandra Tappini, Stephen Wismath
Published in: WALCOM: Algorithms and Computation
Publisher: Springer International Publishing
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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 \(n \ge 12\) vertices admits a 1-planar packing, while such a packing does not exist if \(n \le 10\).