Degree Bounds for Constrained Pseudo-Triangulations Oswin Aichholzer∗ Michael Hoffmann† Bettina Speckmann‡ Csaba D. T´oth§ Abstract called constrained edges.Anedgee/∈ E connecting two points of S is called admissible with respect to G,ifit We introduce the concept of a constrained pointed does not properly cross any edge e ∈ E and if E ∪{e} is pseudo-triangulation TG of a point set S with respect pointed. From now on we will refer to a pointed pseudo- to a pointed planar straight line graph G =(S, E). For triangulation simply as pseudo-triangulation. the case that G forms a simple polygon P with vertex Streinu showed [12] that any pointed set of edges set S we give tight bounds on the vertex degree of TG. can be extended to a pseudo-triangulation by greedily 1 Introduction inserting admissible edges. This implies that a con- strained pseudo-triangulation TG of S exists for every Pseudo-triangulations, also called geodesic triangula- pointed planar straight line graph G and immediately tions, are planar partitions that have received consider- leads to the following question: For which classes of able attention during the last years due to their applica- graphs is the maximum vertex degree of a constrained tions to visibility [8, 9], ray shooting [3, 4], kinetic colli- pseudo-triangulation bounded by a constant? sion detection [1, 6, 7], rigidity [12], and guarding [11]. Here we start to answer this question by giving tight A pseudo-triangle is a planar polygon that has ex- degree bounds for constraining graphs that form a sim- actly three convex vertices, called corners,withinter- ple polygon P with vertex set S. We begin by describ- π nal angles less than . Three concave chains, called side ing a recursive construction of a pseudo-triangulation chains, join the three corners. A pseudo-triangulation of the closed interior of P with maximum vertex de- S n for a set of points in the plane is a partition of the gree five. In contrast, we show that the degree of the S convex hull of into pseudo-triangles whose vertex set dual graph of any pseudo-triangulation of the interior S is exactly . A vertex is called pointed if it has an adja- of a simple polygon cannot be bounded by a constant π cent angle greater than . A planar straight line graph (Section 2.4). Finally, in Section 3 the results are ex- is pointed if every vertex is pointed. tended to construct a constrained pseudo-triangulation Since a pseudo-triangulation is a planar graph, we can of S with maximum vertex degree seven. borrow graph terminology: The degree of a vertex is the number of edges incident to it. Even though a standard 2 The interior of a simple polygon O triangulation has average vertex degree (1), there are In this section we present a recursive construction to n sets of points in the plane for which each possible tri- pseudo-triangulate the closed interior Int (P )ofasim- n − angulation has a vertex of degree 1. In contrast, ple polygon P . At each step we apply one of two oper- Kettner et al. [5] recently established that every point ations to P to obtain polygons of smaller size. The first set in the plane has a pointed pseudo-triangulation of operation prunes aconvexvertexuc from P and either maximum vertex degree five, and this bound is tight. A reduces the size of P by at least one vertex or splits natural question to ask is whether a bound on the max- P in two or more subpolygons. The second operation imum vertex degree is also attainable if certain edges wraps a pseudo-triangle around a reflex vertex ur and are constrained to be part of the pseudo-triangulation. splits P into at least two subpolygons. The vertex ur Assume that we are given a pointed planar straight can reappear only as a convex vertex in at most one of line graph G =(S, E). A constrained pointed pseudo- the resulting subpolygons. Throughout this paper we triangulation TG of S with respect to G is a partition denote a convex vertex with the subscript c and a reflex of the convex hull of S into pseudo-triangles such that vertex with the subscript r. E each edge from is part of the pseudo-triangulation and We define the load of a vertex v, denoted by l(v), to S 1 E each vertex in remains pointed .Theedgesin are be the degree of v minus its constrained degree where ∗ the constrained degree dG(v)ofavertexv denotes its Institute for Software Technology, Graz University of Tech- G nology, [email protected] degree in the constraining graph .Notethatdegree †Institute for Theoretical Computer Science, ETH Z¨urich, always refers to the degree of a vertex with respect to [email protected] the current version of the pseudo-triangulation that we ‡Department of Computing Science, TU Eindhoven, speckman are building. If G is a simple polygon then dG(v)equals @win.tue.nl v ∈ S u, v P §Department of Computer Science, University of California at two for all . We say that an edge ( )of is Santa Barbara, [email protected]. loaded if both u and v have a load greater than zero. 1Rote et al. [10] use the term constrained pseudo-triangulation We maintain the following invariants for all simple to denote a pseudo-triangulation that is a subset of a given trian- polygons P that arise as recursive subproblems: gulation of a point set. 1 Invariant 1 Exactly one edge of P is loaded. 2.1 Prune. Assume that we are given a simple poly- P Invariant 2 The loaded edge is of one of four types: gon that satisfies both invariants and has a loaded edge of either type I or II. I (uc,vc) with l(uc) ≤ 3 and l(vc) ≤ 2. Typ e I. P has a loaded edge (uc,vc) connecting two II (uc,vr) with l(uc) ≤ 3 and l(vr) ≤ 1. convex vertices with l(uc) ≤ 3andl(vc) ≤ 2. Let wc III (uc,vr) with l(uc) ≤ 2 and l(vr) ≤ 2. denote the first convex vertex we encounter when walk- ing away from uc on P not passing over vc. We prune IV (ur,vr) with l(ur) ≤ 2 and l(vr) ≤ 1. uc by adding the pseudo-triangle T (wc,uc,vc)toour In order to establish the invariants for the initial poly- pseudo-triangulation. Note that in this case we always gon P , we pick an arbitrary edge of P and declare it have uc = t(uc)andvc = t(vc). loaded. u Let us now assume that we are given a simple poly- c vc u c v +2 +3 gon P that satisfies both invariants. The appropriate c +3 t(w ) +2 operation to choose depends on the type of the loaded +3 c +3 t(wc) edge e. We distinguish two cases: If e is of type I or II +1 P1 e +1 P2 then we prune, if is of type III or IV then we wrap. P P3 The following sections explain the two operations in detail and illustrate how the invariants are maintained. First, we introduce some additional notation. Recall (a) (b) that a geodesic path between two points inside a sim- ple polygon is the shortest path that connects them Figure 2: A loaded edge of type I: pruning uc. and stays completely within the polygon. The region If the side chain connecting t(wc)tovc consists of a bounded by the three geodesic paths connecting three single diagonal (see Figure 2(a)), then the polygon P vertices a, b,andc is called a geodesic triangle [3]. It formed by pruning uc from P has exactly one loaded consists of a central pseudo-triangle T (a, b, c)andthree edge of either type I (if t(wc) is convex) or type II (if (possibly empty) tails connecting the corners t(a), t(b), t(wc) is reflex). and t(c)ofT (a, b, c) with the vertices a, b,andc (see If the side chain connecting t(wc)tovc consists of sev- Figure 1). eral diagonals, then each polygon Pi formed by pruning Each operation selects a specific pseudo-triangle uc from P has exactly one loaded edge (see Figure 2(b)). T (a, b, c) ⊂ P such that E and the edges of T (a, b, c) Each vertex that newly appears on the side chain be- form a pointed edge set. The set P \ T (a, b, c) is possi- longs to two subpolygons and is convex in both of them. bly split into several (smaller) simple polygons which we To ensure that their total load will not exceed three, we process recursively. The side chains of T (a, b, c) consist consider each of these vertices to have load two in one of constrained edges and diagonals. The load of every subpolygon and load three in the other one. We al- vertex that appears on a side chain is raised by either ternate the loads assigned to each of these vertices as one or two. Furthermore, these vertices also belong to depicted in Figure 2(b) (the numbers inside each sub- either one or two subproblems. Since we are interested polygon indicate the load). This guarantees that P1 (the in an upper bound on the maximal load of any vertex, subpolygon containing vc) has a loaded edge of type I, we assume in the following that every vertex that ap- Pk (the subpolygon containing t(wc)) has a loaded edge pears on a side chain picks up a load of two and belongs of either type I (if t(wc) is convex) or type II (if t(wc) to two subproblems.