Open Problems in the Combinatorics of Visibility and Illumination

Abstract

1991 Primary 52C99. Visibility, illumination, visibility graphs, computational geometry. Supported by NSF grant CCR-9421670. URL: . The \art gallery theorem," that 3 guards su ce and are sometimes necessary to visually cover a polygon of vertices [ ] [ ], typi es the interplay between geometry and combinatorics in a wide variety of subsequent results. The open problems in this collection are intellectual kins to these art gallery-like results. The problems represent a personal selection from the frontier of research on the combinatorics of visibility; no attempt is made to be comprehensive. Updates will be maintained on the Web. In general, a point can , is to, or a point if the line segment is not obstructed. (The metaphors of \visibility" and \illumination" are used interchangeably.) Often this means that cannot contain any point of a set of obstacles, but sometimes is permitted to contain boundary points of the obstacles (line-of-sight grazing contact). Other nuances to the de nition of visibility are appropriate for certain problems, and will be detailed below. Throughout will be used to denote the constant of proportionality for quantities of the form + for some constant , where is the primary input variable (e.g., the number of vertices). Precise fractional bounds will be supplemented by their approximate decimal equivalents.