  The Las Vergnas polynomial is an extension of the Tutte polynomial to
cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as
special case of his Tutte polynomial of a morphism of matroids. While the
general Tutte polynomial of a morphism of matroids has a complete set of
deletion-contraction relations, its specialisation to cellularly embedded
graphs does not. Here we extend the Las Vergnas polynomial to graphs in
pseudo-surfaces. We show that in this setting we can define deletion and
contraction for embedded graphs consistently with the deletion and contraction
of the underlying matroid perspective, thus yielding a version of the Las
Vergnas polynomial with complete recursive definition. This also enables us to
obtain a deeper understanding of the relationships among the Las Vergnas
polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial. We
also take this opportunity to extend some of Las Vergnas' results on Eulerian
circuits from graphs in surfaces of low genus to surfaces of arbitrary genus.
