1. W. D. Grover TRLabs & University of Alberta © Wayne D. Grover 2002, 2003 Graph theory and routing (initial background) E E 681 - Lecture 4

2. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 2 Graph terminology / concepts A graph consists of a finite set of vertices (“nodes”) and a set of edges (“spans”), such that each edge joins a pair of nodes. Nodes are adjacent if they are joined by an edge. Such an edge is incident on the nodes it connects. A graph is simple if it has no parallel edges or self-loops.

3. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 3 Graph terminology / concepts A graph with parallel edges is called a multigraph. A graph in which a number is associated with every edge is called a weighted graph or “capacitated”. An edge is directed if its vertices are an ordered pair. The number of spans incident on a vertex is called the degree of the node.

4. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 4 Graph terminology / concepts A path is a connected edge sequence in which all edges are distinct. If all vertices are also distinct (except possibly the origin and terminus vertices) the path is called simple. If the vertices in a cycle are distinct it is called a simple cycle or circuit. (Note this implies that all edges in the sequence of the walk are also distinct.)

5. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 5 Route - ~ geographical path - ~ signal over a route span - ~ transmission systems link - ~ unit bandwidth unit hop Need to be attentive to these terms for precise communication Routing and flows in transport networks

6. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 6 Hamiltonian and Eulerian Cycles A cycle that connects all of nodes once, is a Hamiltonian cycle. A cycle that traverses all edges once (but may revisit nodes) is an Eulerian cycle.

7. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 7 Hamiltonian Cycles the question: “does graph G contain a Hamiltonian cycle?” is an NP-complete decision problem. –Hamiltonians are more likely in highly connected networks. –any graph of N nodes where every node has degree > N/2 is Hamiltonian Q. What would be the significance for transport network design?

8. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 8 An Eulerian cycle exists (and requires that) every node have even degree Q. What would be the significance for transport network design? Eulerian Cycles

9. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 9 Isomorphism and Homeomorphism Graphs G and H are isomorphic if a 1:1 correspondence can be constructed between nodes in G and H for which: if there is an edge between two nodes in G then there is also an edge between the corresponding nodes in H. (i.e., topological equivalence ) Graphs G and H are homeomorphic if they can be isomorphic only by adding or deleting degree-2 nodes from edges. –relevance to mesh network design and ring-mesh hybrids.

10. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 10 Homeomorphism

11. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 11 Planarity

12. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 12 |E| = 9 |V| = 5  |F| = 6 |F| = |E| - |V| + 2 Planarity - “Euler’s Formula” A connected planar graph divides the plane into non- overlapping faces, some bounded (i.e., those interior to the graph) and some unbounded (at an outside edge of the graph), the number of faces |F| is related to the number of nodes |V| and edges |E| of any planar graph:

13. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 13 A graph GA tree on GA spanning tree on G Trees, spanning trees Q. Significance –to data communications ? –to survivability ?

14. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 14 Two-connected (has a “bridge” node) or “articulation point” This graph that is not connected Two-connected and bi-connected graphs Bi-connectedconnected

15. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 15 Incidence matrix Adjacency list: node Contains a “1” where edge exists “0” otherwise Q. Directed vs. Undirected graphs ? node e.g. “snif” file format distance(etc.) List contains entries only for edges that exist Data structures for representing graphs

16. E E 681 - Lecture 4 © Wayne D. Grover 2002, 2003 16 “disjoint” = fully disjoint (nodes and spans) “span disjoint” = may have common nodes “distinct” = just different in at least one detail Distinct and disjoint routes