A Nebesk‎-Type Characterization for Relative Maximum Genus

This paper concerns the maximum genus orientable surface upon which a given graph cellularly embeds. Classical theorems of Xuong and Nebesk‎ give exact values for the maximum genus. The former is suited to constructing embeddings while the latter is suited to forbidding embeddings of larger genus. However, using either theorem alone requires an exhaustive search to establish the exact value. Herein we examine relative embeddings of graphs, where certain facial cycles and their orientations have been prescribed. The relative graph analogue of Xuongʹs theorem is known. In this paper we establish the relative graph analogue of Nebesk‎ʹs theorem.