We present sweeping line graphs, a generalization of $\Theta$-graphs. We show that these graphs are spanners of the complete graph, as well as of the visibility graph when line segment constraints or polygonal obstacles are considered. Our proofs use general inductive arguments to make the step to the constrained setting. These same arguments could apply to other spanner constructions in the unconstrained setting, removing the need to find separate proofs that they are spanning in the constrained and polygonal obstacle settings.

翻译：我们提出扫描线图（sweeping line graphs），这是$\Theta$图的一种推广。我们证明这些图不仅是完全图的生成子图，而且在考虑线段约束或多边形障碍物时，也是可见性图的生成子图。我们的证明采用通用归纳论证方法，使得能够推广到约束场景。这些论证方法同样适用于无约束场景下的其他生成器构造，从而无需为约束场景和多边形障碍物场景分别寻找独立的生成性证明。