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Stretchability of star-like pseudo-visibility graphs

We present advances on the open problem of characterizing vertex-edge visibility graphs (ve-graphs), reduced by results of O’Rourke and Streinu to a stretchability question for pseudo-polygons. We introduce star-like pseudo-polygons as a special subclass containing all the known instances of non-stretchable pseudopolygons. We give a complete cdmbinatorial characterization and a linear-time decision procedure for star-like pseudo-polygon stretchability and star-like ve-graph recognition. To the best of our knowledge, this is the first problem in computational geometry for which a combinatorial characterization was found by first isolating the oriented matroid substructure and then separately solving the stretchability question. It is also the fist class (as opposed to isolated examples) of oriented matroids for which an efficient stretchability decision procedure based on combinatorial criteria is given. The difficulty of the general stretchability problem implied by Mngv’s Universality Theorem makes this a result of independent interest in the theory of oriented matroids.