  Let $P$ be a set of $n$ points in $\mathrm{R}^2$, and let $\mathrm{DT}(P)$
denote its Euclidean Delaunay triangulation. We introduce the notion of an edge
of $\mathrm{DT}(P)$ being {\it stable}. Defined in terms of a parameter
$\alpha>0$, a Delaunay edge $pq$ is called $\alpha$-stable, if the (equal)
angles at which $p$ and $q$ see the corresponding Voronoi edge $e_{pq}$ are at
least $\alpha$. A subgraph $G$ of $\mathrm{DT}(P)$ is called {\it $(c\alpha,
\alpha)$-stable Delaunay graph} ($\mathrm{SDG}$ in short), for some constant $c
\ge 1$, if every edge in $G$ is $\alpha$-stable and every $c\alpha$-stable of
$\mathrm{DT}(P)$ is in $G$.
  We show that if an edge is stable in the Euclidean Delaunay triangulation of
$P$, then it is also a stable edge, though for a different value of $\alpha$,
in the Delaunay triangulation of $P$ under any convex distance function that is
sufficiently close to the Euclidean norm, and vice-versa. In particular, a
$6\alpha$-stable edge in $\mathrm{DT}(P)$ is $\alpha$-stable in the Delaunay
triangulation under the distance function induced by a regular $k$-gon for $k
\ge 2\pi/\alpha$, and vice-versa. Exploiting this relationship and the analysis
in~\cite{polydel}, we present a linear-size kinetic data structure (KDS) for
maintaining an $(8\alpha,\alpha)$-$\mathrm{SDG}$ as the points of $P$ move. If
the points move along algebraic trajectories of bounded degree, the KDS
processes nearly quadratic events during the motion, each of which can
processed in $O(\log n)$ time. Finally, we show that a number of useful
properties of $\mathrm{DT}(P)$ are retained by $\mathrm{SDG}$ of $P$.
