  The degree/diameter problem for mixed graphs asks for the largest possible
order of a mixed graph with given diameter and degree parameters. Similarly the
\emph{degree/geodecity} problem concerns the smallest order of a $k$-geodetic
mixed graph with given minimum undirected and directed degrees; this is a
generalisation of the classical degree/girth problem. In this paper we present
new bounds on the order of mixed graphs with given diameter or geodetic girth
and exhibit new examples of directed and mixed geodetic cages. In particular,
we show that any $k$-geodetic mixed graph with excess one must have geodetic
girth two and be totally regular, thereby proving an earlier conjecture of the
authors.
