CCCG 2017, Ottawa, Ontario, July 26–28, 2017 Planarity Preserving Augmentation of Topological and Geometric Plane Graphs to Meet Parity Constraints I.Aldana-Galv´an ∗ J.L. ´ Alvarez-Rebollar † J.C. Catana-Salazar ∗ E. Sol´ ıs-Villarreal ∗ J. Urrutia ‡ C. Velarde § Abstract We introduce the augmentation problem to meet parity constraints in topological and plane geometric graphs. We show a family of plane topological graphs such that any augmentation leaves at least 2n 5 vertices without meeting their parity constraints, and a family of plane geometric trees such that any augmentation leaves at least � n 10 � vertices without meeting their parity con- straints. We prove that the problem of adding a min- imum number of edges to plane topological graphs is NP -Hard. When the input graph is a topological tree finding a minimum set of edges that needed to be added to meet a parity constraint is solvable in O(n) time and O(1) space. We also establish a lower bound of � 11n 15 � on the number of necessary edges to augment a topolog- ical graph when the graph is augmentable, and a lower bound of � 6n 11 � on the number of necessary edges to aug- ment a geometric tree when the tree is also augmentable to meet the parity constraints. 1 Introduction A topological graph is a graph together with an embed- ding on the plane, such that the vertices are represented by distinct points and the edges are represented by Jor- dan arcs connecting pairs of vertices. A geometric graph is a graph in which its vertices are represented by points on the plane, and its edges by straight line segments joining pairs of vertices. A planar graph is a graph that can be embedded in the plane in such a way that its edges may intersect only at their endpoints. Such an embedding is called a planar embedding of the graph. * Posgrado en Ciencia e Ingenier´ ıa de la Computaci´on, Universidad Nacional Aut´ onoma de M´ exico, Ciudad de M´ exico, M´ exico, ialdana@ciencias.unam.mx, {j.catanas, solis e}@uxmcc2.iimas.unam.mx † Posgrado en Ciencias Matem´ aticas, Universidad Na- cional Aut´onoma de M´ exico, Ciudad de M´ exico, M´ exico, chepomich1306@gmail.com ‡ Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´ exico, Ciudad de M´ exico, M´ exico, urrutia@matem.unam.mx § Instituto de Investigaciones en Matem´ aticas Aplicadas y en Sistemas, Universidad Nacional Aut´onoma de M´ exico, Ciudad de M´ exico, M´ exico, velarde@unam.mx A plane graph is a planar emmbeding of a planar graph, and we refer to its points as vertices and lines as edges. Two or more geometric graphs are compatible if their union is a plane geometric graph. Given a plane topological (resp. geometric) graph G = (V,E) and a set of parity constraints C = {c 1 ,c 2 , ..., c n } where each v i ∈ V has assigned the con- straint c i (to be of degree odd or to be of degree even), the augmentation problem to meet parity constraints is that of finding a set of edges E � , where E � ∩ E = ∅, such that: 1. G � =(V,E ∪ E � ) is a plane topological (resp. geo- metric) simple graph. 2. The degree of each vertex v i ∈ G � meets its parity constraint c i . Observe that if a vertex of G does not meet its par- ity constraint, then its degree must increase by an odd integer. In what follows we will denote by P the set of vertices of G that do not satisfy its degree constraints in C = {c 1 ,c 2 , ..., c n }. Let H be the graph with vertex set V , and edge set E � . The degree of each vertex in H is odd, and thus H has an even number of vertices. We say that the neighborhood of a vertex is saturated if there is no edge that can be added to G, incident to v, and avoiding edge crossings. For example, if G is a planar graph and the subgraph induced by v i and its neighbors is a wheel with no other vertices inside it, then the neighborhood of v i is saturated, or for short v i is saturated. Thus from now on we will assume that the degree of any vertex in P is smaller than n - 1 and its neighborhood is not saturated. It is easy to see that there are many planar graphs that cannot be extended to meet a set of parity con- straints. For example take a planar graph that is a tri- angulation Δ minus two edges e =(u, v) and e � =(x, y) such that u, v, x, and y are different vertices. Then we cannot change the parities of u and x without breaking the planarity of Δ. Then, the graphs we study in this paper must have the following properties: 1. The graphs are simple. 107