TY - JOUR

T1 - Plane graphs with parity constraints

AU - Aichholzer, O.

AU - Hackl, T.

AU - Hoffmann, M.

AU - Pilz, A.

AU - Rote, G.

AU - Speckmann, B.

AU - Vogtenhuber, B.

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

U2 - 10.1007/s00373-012-1247-y

DO - 10.1007/s00373-012-1247-y

M3 - Article

VL - 30

SP - 47

EP - 69

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -