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We present improved upper bounds on the spanning ratio of constrained$\theta$-graphs with at least 6 cones and constrained Yao-graphs with 5 or atleast 7 cones. Given a set of points in the plane, a Yao-graph partitions theplane around each vertex into $m$ disjoint cones, each having aperture $\theta= 2 \pi/m$, and adds an edge to the closest vertex in each cone. ConstrainedYao-graphs have the additional property that no edge properly intersects any ofthe given line segment constraints. Constrained $\theta$-graphs are similar toconstrained Yao-graphs, but use a different method to determine the closestvertex.  We present tight bounds on the spanning ratio of a large family ofconstrained $\theta$-graphs. We show that constrained $\theta$-graphs with $4k+ 2$ ($k \geq 1$ and integer) cones have a tight spanning ratio of $1 + 2\sin(\theta/2)$, where $\theta$ is $2 \pi / (4k + 2)$. We also present improvedupper bounds on the spanning ratio of the other families of constrained$\theta$-graphs. These bounds match the current upper bounds in theunconstrained setting.  We also show that constrained Yao-graphs with an even number of cones ($m\geq 8$) have spanning ratio at most $1 / ( 1 - 2 \sin (\theta/2) )$ andconstrained Yao-graphs with an odd number of cones ($m \geq 5$) have spanningratio at most $1 / ( 1 - 2 \sin (3\theta/8) )$. As is the case with constrained$\theta$-graphs, these bounds match the current upper bounds in theunconstrained setting, which implies that like in the unconstrained settingusing more cones can make the spanning ratio worse.

Spanning Properties of Yao and $\Theta$-Graphs in the Presence of Constraints