  In this thesis, we study two different graph problems.
  The first problem revolves around geometric spanners. Here, we have a set of
points in the plane and we want to connect them with straight line segments,
such that there is a path between each pair of points that does not make a
large detour. If we achieve this, the resulting graph is called a spanner. We
focus our attention on $\Theta$-graphs, which are constructed by connecting
each point with its nearest neighbour in a fixed number of cones. Although this
construction is very straight-forward, it has proven challenging to fully
determine the properties of the resulting graphs. We show that if the
construction uses 5 cones, the resulting graphs are still spanners. This was
the only number of cones for which this question remained unanswered. We also
present a routing strategy on the half-$\Theta_6$-graph, a variant of the graph
with 6 cones. We show that our routing strategy finds a path whose length is at
most a constant factor from the straight-line distance between the endpoints.
Moreover, we show that this routing strategy is optimal.
  In the second part, we turn our attention to flips in triangulations. A flip
is a simple operation that transforms one triangulation into another. It turns
out that with enough flips, we can transform any triangulation into any other.
But how many flips is enough? We present an improved upper bound of $5.2n -
33.6$ on the maximum flip distance between any pair of triangulations with n
vertices. Along the way, we prove matching lower bounds on each step in the
current algorithm, including a tight bound of $(3n - 9)/5$ flips needed to make
a triangulation 4-connected. In addition, we prove tight $\Theta(n \log n)$
bounds on the number of flips required in several settings where the edges have
unique labels.
