Kuratowski-Type Theorems for Average Genus

Graphs of small average genus are characterized. In particular, a Kuratowski-type theorem is obtained: except for finitely many graphs, a cutedge-free graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list or those exceptions. A Kuratowski-type theorem for graphs of maximum genus 1 is also given. Some of the methods used in obtaining these results involve variations of a classical result of Whitney.