Let S be a set of n points in general position in the plane. Together with S we are given a set of parity constraints, that is, every point of S is labeled either even or odd. A graph G on S satisfies the parity constraint of a point p¿¿¿S, if the parity of the degree of p in G matches its label. In this paper we study how well various classes of planar graphs can satisfy arbitrary parity constraints. Specifically, we show that we can always find a plane tree, a two-connected outerplanar graph, or a pointed pseudo-triangulation which satisfy all but at most three parity constraints. With triangulations we can satisfy about 2/3 of all parity constraints. In contrast, for a given simple polygon H with polygonal holes on S, we show that it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
|Title of host publication||Algorithms and Data Structures (Proceedings 11th International Workshop, WADS 2009, Banff, Alberta, Canada, August 21-23, 2009)|
|Editors||F. Dehne, M. Gavrilova, J.-R. Sack, C.D. Tóth|
|Place of Publication||Berlin|
|Publication status||Published - 2009|
|Name||Lecture Notes in Computer Science|