  A degree monotone path in a graph $G$ is a path $P$ such that the sequence of
degrees of the vertices in the order in which they appear on $P$ is monotonic.
The length of the longest degree monotone path in $G$ is denoted by $mp(G)$.
This parameter, inspired by the well-known Erdos-Szekeres theorem, has been
studied by the authors in two earlier papers. Here we consider a saturation
problem for the parameter $mp(G)$. We call $G$ saturated if, for every edge $e$
added to $G$, $mp(G+e) >mp(G)$, and we define $h(n,k)$ to be the least possible
number of edges in a saturated graph $G$ on $n$ vertices with $mp(G) < k$,
while $mp(G+e) \geq k$ for every new edge $e$.
  We obtain linear lower and upper bounds for $h(n,k)$, we determine exactly
the values of $h(n,k)$ for $k=3$ and $4$, and we present constructions of
saturated graphs.
