Graph genus most commonly refers to the orientable genus 
of a graph 
, namely the minimum genus of an
orientable surface in which 
can be embedded (Mohar
1998, Wan 2023).
When orientability is not understood from context, the qualifier is needed. The nonorientable genus 
instead minimizes the genus
of a nonorientable surface in which 
can be embedded.
The Euler genus of 
is 
and treats the orientable and nonorientable cases in a common scale.
Deciding whether a graph has orientable genus at most 
is an NP-complete problem when 
is part of the input (Thomassen 1989). For each fixed 
, however, a graph
embedding in the orientable surface of
genus 
can be found, or its nonexistence established, in linear time
(Mohar 1999).
