We introduce and study the 1-planar packing problem: Given k graphs with n vertices G1,…,Gk, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each Gi 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.
Packing Trees into 1-Planar Graphs
De Luca, Felice;Di Giacomo, Emilio;Liotta, Giuseppe;Tappini, Alessandra;
2020
Abstract
We introduce and study the 1-planar packing problem: Given k graphs with n vertices G1,…,Gk, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each Gi 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
