Henneberg construction

Transcription

1 Henneberg construction Seminar über Algorithmen FU-Berlin, WS 2007/08 Andrei Haralevich Abstract: In this work will be explained two different types of steps of Henneberg construction. And how Henneberg construction works for embedding any Laman graph as pointed Pseudo-Triangulation. Introduction: Def.: Pointed Vertex is Vertex that is adjacent to an angle larger than π Fig. 1: Pointed Vertex Fig. 2: Non-pointed Vertex Def.: Pseudo-Triangle is a simple polygon with exactly three convex vertices (and all the others reflex). Fig. 3: Pseudo-Triangles Def.: Pseudo-Triangulation is a partition of a region of the plane into Pseudo-Triangles. Def.: Pointed Pseudo-Triangulation is a Pseudo-Triangulation of a convex polygon in which every vertex is pointed vertex. Fig. 4: Pointed Pseudo-Triangulation Fig. 5: Non-pointed Pseudo- Triangulation Def.: Laman graph. A graph G with n vertices and m edges is a Laman graph if m = 2n 3 and every subset of k vertices spans at most 2k 3 edges. Fig. 6: Laman Graph Laman graphs are the fundamental objects in 2-dimensional Rigidity Theory. Also known as isostatic or generically minimally rigid graphs.

2 1. Henneberg construction: - Henneberg Step 1 (vertex addition): the new vertex is connected via two new edges to two old vertices. - Henneberg Step 2 (edge splitting): a new vertex is added on some edge (thus splitting the edge into two new edges) and then connected to a third old vertex. Equivalently, this can be seen as removing an edge, then adding a new vertex connected to its two endpoints and to some other vertex. Fig. 7: Henneberg Construction. Top row, when the new vertex is added on the outside face. Bottom row, when it is added inside a pseudo-triangular face. 2. Drawing of pseudo-triangulations with Henneberg construction: Lemma: A Laman graph has a Henneberg construction starting from any prescribed subset of two vertices. Proof: If the number of edges in a graph is 2n - 3, there exists at least one vertex of degree 2 or 3. Otherwise, if all nodes have degree at least 4, we have at least 4n/2 = 2n edges, what violates Laman condition. If there is a degree 2 vertex in the graph, remove the vertex and its adjacent edges (Henneberg I Step in reverse). If there is a degree 3 vertex, remove that vertex and consider its 3 neighbors a; b; c. These cannot form a triangle, since it would violate the Laman condition for k = 4 with nodes a; b; c; x. Put the missing edge among these 3 vertices. (Henneberg II Step in reverse). Inductively we can proceed with removing vertices from graph without violating Laman condition, till we get only two vertices connected with a single edge.

3 Lemma 1: (Fixing the Outer Face) Embedding of a plane Laman graph as a pseudo-triangulation reduces to the case when the outer face is a triangle. Fig. 8: Laman graph G in a triangle. Proof: Let G be a plane Laman graph with an outer face having more than three vertices. We construct another Laman graph G of n+3 vertices by adding 3 vertices in the outer face and connecting them to a triangle containing the original graph in its interior. Then we add an edge from each of the 3 new vertices to three distinct vertices on the exterior face of G. These new 3 edges can always be add so that we are splitting face between G and G into 3 pseudo-triangles, i.e. all points in outer face of G remains pointed. We now realize G as a pseudo-triangulation with the new triangle as the outer face. Def: The feasibility region of an arbitrary point p on the boundary of face F is the wedge-like region inside F from where tangents to the boundary of F at p can be taken. Note: The feasibility region of several points is the intersection of their feasibility regions. Lemma 2: (geometric lemma): Every plane Laman graph G can be embedded as a pseudotriangulation. Proof: Let G n be a plane Laman graph on n vertices vertices with a triangular outer face (according to Lemma 1). Assume we have a plane Henneberg construction for G starting with the outer face F and adding vertices only on interior faces (this is just a technical simplification reducing the size of our case analysis. The Henneberg steps would work just as well for insertions on the outer face.). Let s show that there always exists a way of placing new vertex of degree 2 or 3 (marked with red color) inside a face which realizes a compatible partitioning of the face into pseudo-triangles as prescribed by the Henneberg steps. In a Henneberg I step, the new vertex v is inserted on an interior face (which is a pseudo-triangle), and joined to two old vertices i and j. The new edges vv i and vv j partition the face F and its three corners into two pseudo-triangles. The following cases may happen (see Fig. 8-12).

4 - neither v i nor v j are the corner of F (1-2 cases) - vertex v i or v j is a corner of F (3-4 cases) - both v i and v j are the a corner of F (5 cases) Fig. 8: Case 1 Fig. 9: Case 2 Fig. 10: Case 3 Fig. 11: Case 4 Fig. 12: Case 5

5 For analysis of a Henneberg 2 step let s illustrate only one representative case: Consider the (embedded) interior face F with four corners obtained by removing an interior edge pipj, and let pk be a vertex on the boundary of F. We must show that there exists a point p inside F such that, when connected to pi, pj and pk partitions it into three pseudo-triangles and is itself pointed. The three line segments ppi, ppj and ppk must be tangent to the side chains of F. An important fact is that the feasibility region of pi and pj always contains the part of the supporting line of the removed edge pipj, and that the feasibility region of any other vertex pk cuts an open segment on it. In fact, the feasibility region of pk intersects the feasibility region of pi and pj in a non-empty feasible 2- dimensional region on one side or the other (or both) of this segment. You can now easily see that this intersection region not only non-empty, but it also it contains a subregion where we can place a new vertex, such that it is pointed and it splits face F into three pseudo-triangles. See Fig. 13. Fig. 13: Case 5 Literature: 1. Ruth Haas, David Orden, Günter Rote, Francisco Santos, Brigitte Servatius, Herman Servatius, Diane Souvaine, Ileana Streinu, and Walter Whiteley. Planar minimally rigid graphs and pseudo-triangulations. Computational Geometry, Theory and Applications 31 (2005), Ileana Streinu. Pseudo-triangulations, rigidity and motion planning. Discrete and Computational Geometry, 34: , 2005.