Plane graphs with parity constraints

O. Aichholzer, T. Hackl, M. Hoffmann, A. Pilz, G. Rote, B. Speckmann, B. Vogtenhuber

Research output: Contribution to journalArticleAcademicpeer-review

3 Citations (Scopus)
101 Downloads (Pure)


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 that satisfy all but at most three parity constraints. For triangulations we can satisfy about 2/3 of the parity constraints and we show that in the worst case there is a linear number of constraints that cannot be fulfilled. In addition, we prove that for a given simple polygon H with polygonal holes on S, it is NP-complete to decide whether there exists a triangulation of H that satisfies all parity constraints.
Original languageEnglish
Pages (from-to)47-69
JournalGraphs and Combinatorics
Issue number1
Publication statusPublished - 2014


Dive into the research topics of 'Plane graphs with parity constraints'. Together they form a unique fingerprint.

Cite this