Henneberg-construction (Streinu) and drawing of pseudotriangulations Seminar über Algorithmen

1. Henneberg-construction (Streinu) and drawing of pseudotriangulations Seminar über Algorithmen FU-Berlin, WS 2007/08 Andrei Haralevich

2. Introduction Def.: Pointed Vertex is Vertex that is adjacent to an angle larger than π Pointed Vertex Non-pointed Vertex

3. Introduction Def.: Pseudo-Triangle is a simple polygon with exactly three convex vertices (and all the others reflex).

4. Introduction 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. Pointed Pseudotriangulation Non-Pointed Pseudotriangulation (Red Vertex is not pointed)

5. 5 5 7 6 1 6 3 7 1 2 8 9 2 4 10 17 16 8 4 11 12 15 10 13 3 14 9 Introduction Def.: Laman graphs. 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. 10-Vertices, 17-Edges n*2 - 3 = 2*10 – 3 = 17 Laman graphs are the fundamental objects in 2-dimensional Rigidity Theory. Also known as isostatic or generically minimally rigid graphs.

6. 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.

7. Henneberg Construction Example of Henneberg Construction forLaman Graph (8-vertices, 13-edges)2*n-3=m; < – > 2*8-3=13;

8. Henneberg Construction Staring with 2 vertices and 1 edge

9. Henneberg Construction Henneberg Type I (3 vertices and 3 edge) Red color – news edges

10. Henneberg Construction Henneberg Type I (3 vertices and 3 edge)

11. Henneberg Construction Henneberg Type I (4 vertices and 5 edge) Red color – news edges

12. Henneberg Construction Henneberg Type I (4 vertices and 5 edge)

13. Henneberg Construction Henneberg Type II (5 vertices and 7 edge) Grey color – old edge Red color – news edges

14. Henneberg Construction Henneberg Type II (5 vertices and 7 edge)

15. Henneberg Construction Henneberg Type II (6 vertices and 9 edge) Grey color – old edge Red color – news edges

16. Henneberg Construction Henneberg Type II (6 vertices and 9 edge)

17. Henneberg Construction Henneberg Type II (7 vertices and 11 edge) Grey color – old edge Red color – news edges

18. Henneberg Construction Henneberg Type II (7 vertices and 11 edge)

19. Henneberg Construction Henneberg Type II (8 vertices and 13 edge) Grey color – old edge Red color – news edges

20. Henneberg Construction Initial Laman Graph is constucted

21. Drawing of pseudotriangulations 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. 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 have now realized G′ as a pseudo-triangulation with the new triangle as the outer face.

22. Drawing of pseudotriangulations Def: The feasibility region of an arbitrary vertex 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. Feasibility Region of Point 1 Feasibility Region of Point 2 Feasibility Region of Point 3 Feasibility Region of Point 4 Note: The feasibility region of several points is the intersection of their feasibility regions.

23. Drawing of pseudotriangulations Drawing of pseudo-triangulations with Henneberg construction Lemma 2: (The Geometric Lemma) Every plane Laman graph G can be embedded as a pointed pseudo-triangulation. Proof: Let Gn be a plane Laman graph on n vertices with a triangular outer face (according to Lemma 1). Assume we have a plane Henneberg construction for graph Gn starting with the outer face and adding vertices only on interior faces (technical simplification reducing the size of our case analysis). Now we want to show that there always exists a way of placing a vertex p (of degree 2 or 3) inside of a face which realizes a compatible partitioning of the face into pseudo-triangles as prescribed by the Henneberg steps.

24. Drawing of pseudotriangulations Henneberg I step • For 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 vvi and vvj partition the face F and its three corners into two pseudo-triangles. • The following cases may happen: • neither vi nor vj are the corners of F (1-2 cases) • vertex vi or vj is a corner of F (3-4 cases) • - both vi and vj are the corner of F (5 cases)

25. Drawing of pseudotriangulations Neither vi nor vj is a corner of F (1-2 cases) Case 1 Case 2

26. Drawing of pseudotriangulations Vertex vi or vj is a corner of F (3-4 cases) Case 3 Case 4

27. Drawing of pseudotriangulations Both vertices vi and vj are a corners of F (5 cases) Case 5

28. Drawing of pseudotriangulations Henneberg II step 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. Feasibility Regions of point i, j, k 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.

29. Drawing of pseudotriangulations We can see that: - the feasibility region of pi and pj always contains the part of the removed edge pipj - the feasibility region of pk intersects the feasibility region of pi and pj in a non-empty feasible region on one side or the other side (or both sides) of this segment. Intersection of their feasibility regions of vertices i, j, k

30. Drawing of pseudotriangulations We 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. Inserting of a new red vertex, that creates pseudotriangulation. So Lemma 2 is geometrically proven!

31. Drawing of pseudotriangulations Thank you for your attention