Embedding and canonizing graphs of bounded genus in logspace

Abstract

Graph embeddings of bounded Euler genus (that means, embeddings with bounded orientable or nonorientable genus) help to design time-efficient algorithms for many graph problems. Since linear-time algorithms are known to compute embeddings of any bounded Euler genus, one can always assume to work with embedded graphs and, thus, obtain fast algorithms for many problems on any class of graphs of bounded Euler genus. Problems on graphs of bounded Euler genus are also important from the perspective of finding space-efficient algorithms, mostly focusing on problems related to finding paths and matchings in graphs. So far, known space-bounded approaches needed the severe assumption that an embedding is given as part of the input since no space-efficient embedding procedure for nonplanar graphs was known. The present work sidesteps this assumption and shows that embeddings of any bounded Euler genus can be computed in deterministic logarithmic space (logspace); allowing to generalize results on the space complexity of path and matching problems from embedded graphs to graphs of bounded Euler genus. The techniques developed for the embedding procedure also allow to compute canonical representations and, thus, solve the isomorphism problem for graphs of bounded Euler genus in logspace.