1. Faculty of Civil Engineering Institute of Construction Informatics, Prof. Dr.-Ing. Scherer Institute of Construction Informatics, Prof. Dr.-Ing. Scherer Technische Universität Dresden GIS 1 Geo Information Systems Part 4 Graphs Prof. Dr.-Ing. Raimar J. Scherer Institute of Construction Informatics Dresden, 05.07.2006

2. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 2 Graphs: definition m Finite graph := G={ V, E } m Contineous (Coherent ) graph := V v i,v j = for those at least one path exists (1) cyclical graphs (2) continuous without cycle: exclusively cyclical: e.g. real estate plane e.g. tree, m planar graph: all intersection points of the edges are vertexes Caution: V vertexes = points but V points ≠ vertexes; that means an edge can be a polygon edges vertex

3. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 3 Example of the graph representation Example of the graph representation of the topology of a real estate plane Example Real estate plane boundary representation 1 2 5 6 4 3 7 8 9 A1 A3 A6 A2 A5 A7 A11 A9 A4 A8 A10 A12 R11R12 R22 R21 0 R11R12R21R22 A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 123456789 object edges vertices

4. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 4 Example: Cube Topology represented as graph S1, S2 – Base and top area S3...S6 – Side area E1...E12 – Edges V1...V8 - Vertices

5. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 5 Incidence Incidence = relationship between elements of different type {edges – vertexes} from = {(a,1),(a,2)} {vertexes - edges} from = {(1,a),(1,b), (1,c),(1,d)} mathematical definition:designate falling together or interleave of the elements of a graph e.g. edge incidences with its start vertex SV and end vertex EV 1 2 b c d a 1 a

6. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 6 Adjacent adjacent = relationship between elements of equal type, where the relationship is defined by another type of element {vertexes : edge} from = a:(1,2) {edges : vertex} from = 1:(a,b,c,d) 1 2 b c d a 1 a

7. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 7 Isomorphic Mapping Mapping where the incidence and adjacent are the same (invariant) It is a special case of a topological mapping

8. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 8 Classification of the possibilities for topological queries 1. Vertex-vertex relationship (a) which vertex V i adjazises with vertex V j ? (b) how many vertexes adjazises vertex V j ? (c) which distance (incidence and adjazize steps) is between vertex V i and vertex V j ? Eample: d=5 Remark: With incidence and adjacent we can define a topological metric

9. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 9 Classification of the possibilities for topological queries 2. Vertex-edges relationship (a) which vertex V i incidences with edge E j ? (b) how many edges incidences vertex V j ? (c) which distance is between vertex V i and edge E j ?

10. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 10 Classification of the possibilities for topological queries 3. Vertex-area relationship (a) which vertex V i provides an island in area S j, i.e. is inside the area. (b) which vertex V i is at the border of area S j ? (c) how many areas incidences vertex V j ?

11. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 11 Classification of the possibilities for topological queries 4. edges-edges relationship (a) which edge E i branches from edge E j ? This question is common by the analysis of trees, i.e. for networks of pipelines or traffic. (b) which edge E i cuts in which vertex V i the edge E j ? (c) which edge E i crosses edge E j ? This question is only answerable with a planar graph if an additional and marked vertex is inserted.

12. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 12 Classification of the possibilities for topological queries 5. edges-area relationship (a) which edge E i ends in area S j ? (b) which edge E i cuts area S j ? (c) which edge E i is border line of area S j ? (d) which edge E i belongs to which area?

13. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 13 Classification of the possibilities for topological queries 5. area-area relationship (a) area S i has which neighbour area S j ? (b) how many areas adjoin with area S j ? (c) which area S j is an island of area S j ?

14. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 14 Representation by Matrices Incident-Matrix B: edges-nodes B (j,i) = 1 if edge j comes from node i B (j,i) = -1if edge j ends in node i else:B (j,i) = 0  Adjacent-Matrix A: nodes-edges A = B T B A (j,j) … number of edges that meet in node j A (j,i) = -1 if connection i,j is an edge else:A (j,i) = 0 Remark: Matrices can become very huge J J i i

15. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 15 Example: Respresentation by Matrices Example: a bc d ef g 125 126 12 43 5 6 edge-node start- node end- node a14 b15 c25 d23 e36 f46 g56 nodes edges                               110000 101000 100100 000110 010010 010001 001001 B nodes                            311100 130011 102001 100210 010120 011002 A Remark: Nodes are numbered according to minimal distance

16. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 16 Topological Relationships and Consistance Checks Notice for further extention