2024/06/06 by Senja Barthel, Barthel, Senja, Fabio Buccoliero +1
Computer Science · Engineering · #05C10 #05C45 #05C62 #57M15 #57M25 #Advanced Materials and Mechanics #Cellular Automata and Applications #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2406.03800
openalex publication_date 2024/06/06 · openalex created_date 2024/06/08 · openalex updated_date 2026/07/28
For a given spatial graph G ⊂ ℝ3, we would like to find a closed orientable surface S embedded in ℝ3 in which G is cellular embedded. However, for general G this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface S can always be found. The argument is based on decomposing G into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of G. We generalize the procedure to an algorithm that, if successful, constructs S for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.