Proximity Constraints and Representable Trees

This paper examines an infinite family of proximity drawings of graphs called open and closed ß-drawings, first defined by Kirkpatrick and Radke [15, 21] in the context of computational morphology. Such proximity drawings include as special cases the well-known Gabriel, relative neighborhood and strip drawings. Complete characterizations of those trees that admit open /^-drawings for 0 < ß < iz^f) and ^f) < ß < °° or closed /^-drawings for 0 < ß < ^J^ and ^E) < ß < oo are given, as well as partial characterizations for other values of ß. For the intervals of ß in which complete characterizations are given, it can be determined in linear time whether a tree admits an open or closed /^-drawing, and, if so, such a drawing can be computed in linear time in the real RAM model. Finally, a complete characterization of all graphs which admit closed strip drawings is given.