Graphs

1. Graphs Rosen 8.1, 8.2

2. There Are Many Uses for Graphs! • Networks • Data organizations • Scene graphs • Geometric simplification • Program structure and processes • Lots of others…. • Also applications (e.g., species structure—phylogeny tree)

3. Definitions • A simple graph G = (V,E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements of V called edges. • A multigraph G = (V,E) consists of a set V of vertices, a set E of edges, and a function f from E to {{u,v} | u,v V, u ≠ v}. The edges e1 and e2 are called multiple or parallel edges if f(e1) = f(e2).

4. Properties Simple graph • Undirected • Single edges • No loops Multigraph • Undirected • Multiple edges • No loops

5. Detroit New York Denver Chicago San Francisco Atlanta Simple Graph (computer backbone) Detroit Los Angeles New York Denver Chicago San Francisco Atlanta MultiGraph (computer backbone with redundant connections) Los Angeles Examples

6. 4 1 7 Digraph for equivalence relation on {1,4,7} Directed Graph • A directed graph consists of a set of vertices V and a set of edges E that are ordered pairs of V • Loops are allowed • Multiple edges are allowed

7. Adjacency and Degree • Two vertices u and v in an undirected graph are called adjacent (or neighbors) in G if e = {u,v} is an edge of G. The edge e is said to connect u and v. The vertices u and v are called the endpoints of e. • The degree of a vertex in an undirected graph is the number of edges that are incident with (or connect) it, except that a loop at a vertex contributes twice to the degree. The degree of the vertex v is denoted by deg(v).

8. Handshaking Theorem Let G = (V,E) be an undirected graph with e edges. Then (Note that this applies even if multiple edges and loops are present.)

9. How many edges are there in a graph with ten vertices each of degree 6? Since the sum of the degrees of the vertices is 6·10 = 60, it follows that 2e =60. Therefore, e = 30.

10. Since deg(v) is even for v  V1, this term is even. Prove that an undirected graph has an even number of vertices of odd degree. Let V1 and V2 be the set of vertices of even degree and the set of vertices of odd degree, respectively, in an undirected graph G = (V,E). Then

11. Prove that an undirected graph has an even number of vertices of odd degree. Furthermore, the sum of these two terms is even, since the sum is 2e. Hence, the second term in the sum is also even. (Why?) Since all the terms in the sum are odd, there must be an even number of such terms. (Why?) Thus there are an even number of vertices of odd degree. What can we say about the vertices of even degree?

12. Cycles The cycle Cn, n≥3, consists of n vertices v1, v2,…, vn, and edges {v1, v2}, {v2, v3},…, {vn-1, vn}, {vn, v1}. C3 C4 C5

13. Tree A circuit is a path of edges that begins and ends at the same vertex. A path or circuit is simple if it does not contain the same edge more than once. • A tree is a connected undirected graph with no simple circuits. • A forest is a set of trees that are not connected.

14. Q Q Q Q Q Q Q Q Q Q Example: Quadtree Q Q Q Q Q Linked Global Quadtrees Q Q Q Q

15. Q Q Q Q Q Q Q Q Q Q Example: Quadtree } Q Q Q Q Q 22 levels Linked Global Quadtrees Q Q Q Q Regular Triangularization (128 x 128 grid points)

16. Quadtree: Terrain • Global Terrain (elevation & imagery) • Multiresolution • Scalable (multiple terabyte databases) • Efficient (100 to1or more reduction)

17. View-Dependent Simplification

18. Base Mesh Original Mesh Building a Vertex Hierarchy Quadric Simplification Vertex Tree

19. Base Mesh Original Mesh Meshing From a Vertex Hierarchy Geometry & Appearance Metric