  Let $P$ be a set of $n$ points in the plane. A geometric graph $G$ on $P$ is
said to be {\it locally Gabriel} if for every edge $(u,v)$ in $G$, the disk
with $u$ and $v$ as diameter does not contain any points of $P$ that are
neighbors of $u$ or $v$ in $G$. A locally Gabriel graph is a generalization of
Gabriel graph and is motivated by applications in wireless networks. Unlike a
Gabriel graph, there is no unique locally Gabriel graph on a given point set
since no edge in a locally Gabriel graph is necessarily included or excluded.
Thus the edge set of the graph can be customized to optimize certain network
parameters depending on the application. In this paper, we show the following
combinatorial bounds on edge complexity and independent sets of locally Gabriel
graphs:
  (i) For any $n$, there exists locally Gabriel graphs with $\Omega(n^{5/4})$
edges. This improves upon the previous best bound of $\Omega(n^{1+\frac{1}{\log
\log n}})$.
  (ii) For various subclasses of convex point sets, we show tight linear bounds
on the maximum edge complexity of locally Gabriel graphs.
  (iii) For any locally Gabriel graph on any $n$ point set, there exists an
independent set of size $\Omega(\sqrt{n}\log n)$.
