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# Graph Theory

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1. Graph Theory CSRU1400, Fall 2007 Ellen Zhang

2. Outline • Introduction to graph • Is there a Eulerian trail, curcuit? • Isomorphism and planarity • Shortest path problem

3. Graph • G=(V,E), V is the set of vertices (nodes), E is the set of edges • Connected graph: where there is a path between any pair of distinct nodes • A graph H is a subgraph of a G if all nodes and edges in H are also nodes and edges in G, i.e. H=(V’,E’) where • A connected component of a graph G is a connected subgraph H of G such that no other connected subgraph of G containing H exists. • Simple graph: no loop or parallel edges c d a b e f V={a, b, c, d, e, f} E={(a,b), (a,c), (a,d),(c,d), (b,d), (e,f)}

4. walk no repeated edge no repeated vertices trail path use every edge closed circuit Eulerian trail use every edge nontrivial & only repeated node is first/last Eulerian circuit cycle use every node Hamiltonian cycle Walks in graph

5. Does a graph have an Eulerian trail ? C • A Walk that go through all edges exactly once • Proposition: in any graph, if a vertex x has an odd degree, then x cannot be an interior vertex (i.e., a vertex other than the starting or stopping point) in an Eulerian trail. • For interior vertex: vs…e1xe2 … e3xe4 … ve D A B

6. What kind of graph has Eulerian circuit ? • Graph G is called Eulerian if G has an Eulerian circuit (a closed walk that used every edge once and only once) • Let G be a connected graph. G is Eulerian if and only if every node in G has even degree. 1 2 3 4 5 6 7 8 10 9

7. Construct Eulerian circuit • Find any circuit C in graph G • starting at a certain node, walking along unused edges until getting back to original node. • Form a graph G’ by removing from G all edges in C and all vertices that have no edges left. • G’ might not be connected • G’ consists of graphs where every node has even degree; • Use same process 1 to determine Eulerian circuits for G’ or its components. • Piece together circuit C, and Euleriancurcuits found for G’, and get an Euleriancurcuit for G. 1 2 3 4 5 6 7 8 10 9

8. Eulerian Trail • A connected graph G has an Eulerian trail if and only if G has exactly two nodes of odd degree. Moveover, the trail must begin and end at these two nodes • If G has exactly two nodes of odd degree, after adding an edge connecting these two nodes, G has Eulerian circuit (by previous theorem). By removing added edge from circuit, we get a trail starting/ending at the two nodes. • If G has an Eulerian trail, it’s easy to see that the starting/ending nodes have odd degree, all other nodes have even degree.

9. Example 1 Does this graph as Eulerian circuit or Eulerian trail ? Find the Eulerian circuit or trail.. 2 5 6 3 4

10. Outline • Introduction to graph • Is there a Eulerian trail, curcuit? • Isomorphism and planarity • Shortest path problem

11. Isomorphic Graphs • Formal Def. Graph G and H are called isomorphic if there is a one-to-one correspondence f, between the vertices in G to vertices in H, so that (u,v) is edge in G if and only if (f(u),f(v)) is edge in H. • One can rearrange G to get H z 3 4 y x u v 1 2 5 6 w

12. Are these two graphs isomorphic ? ƒ(a) = 1 ƒ(b) = 6 ƒ(c) = 8 ƒ(d) = 3 ƒ(g) = 5 ƒ(h) = 2 ƒ(i) = 4 ƒ(j) = 7

13. Checking isomorphism • Two graphs that are isomorphic to one another must have • Same number of nodes • Same number of edges • Same number of nodes of any given degree • Same number of cycles • Same number of cycles of any given size • A pair of graphs satisfying above conditions are not necessarily isomorphic

14. Examples

15. Planar Graph • Often we want to draw graphs as clean as possible, as few crossings as possible • Can we connect three factories to three utilities without one service crossing over the other ? Three utilities Three factories

16. Definitions • A graph is called planar if it can be drawn on a plane so that no edges cross. Such a drawing is called an embedding. • Bipartite graph: the set of nodes can be partitioned into two sets so that every edge has one endpoint in one set, one endpoint in another set. • Complete bipartite graph, Km,n, is a bipartite graph with nodes S1={a1,…,am} and S2={b1,…,bn}, and every nodes in S1 is connecting to every node in S2. • Complete graph, Kn

17. Are these graphs planar? • K4 • K3,2 • Kn,2

18. Are these graphs planar? • K5 • K3,3

19. Kuratowski’s Theorem • A graph G is planar if and only if contains no “copies” of K3,3 or K5 as subgraphs • A subdivision of a graph results from inserting vertices into edges • More strictly speaking, a graph is planar if and only if it does not contain a subgraph that is a subdivision of K5 or K3,3.

20. Practice Problem • Are the following graphs planar ?

21. Euler’s Formula for planar graph • Def: for a planar graph G embedded in the plane, a face of the graph is a region of the plane created by the drawing. Note that we have one unbounded region. K4 a d b b

22. Euler’s Formula for Planar Graph • Basic facts: Given an embedding of a graph, • Removing any edge from a cycle decrease by 1 the num. of faces • Every edge on a cycle borders exactly two faces • For acyclic graph, # of faces is 1 • Euler’s Formula: for any connected planar graph G embedded in the plane with V vertices, E edges and F faces, we have V+F=E+2. • Can be proved (see the book) by induction.

23. Eular’s Formula • Also holds for polyhedrons E=12,V=8,F=6 E=6 V=4 F=4

24. Polyhedra represented as Planar Graph Pick one face and stretch it out…

25. Now why K3,3 is not planar ? • Prove by contradiction. • If K3,3 is planar, then • Since V=6, E=9, by Euler’s formula, V+F=E+2, F=5 • But by another theorem, we have E>=2*F, i.e, F<=4.5. (See book P.542-543 for the detail). • They cannot be both true, so our assumption (K3,3 is planar) is not true. • So K3,3 is not planar

26. Outline • Introduction to graph • Is there a Eulerian trail, curcuit? • Isomorphism and planarity • Shortest path problem

27. Shortest Path Problem • Given a weighted graph and two vertices, u and v, find a path with minimum total weights between u and v • Applications: driving direction, flight reservation etc

28. Dijkstra’s Algorithm • Find the shortest paths from a single source to all other nodes, for weighted graph where all weights are non-negative • Ideas: • Maintain a set S of vertices whose final shortest-path weights from source have been decided • Select from V-S, a vertex with minimum shortest-path estimate, x, insert into S • Do we get better paths now by going through x ? If so, updating shortest-path estimates for nodes in V-S • Repeat until S=V, i.e., shortest path to all nodes have been determined

29. Idea of Dijkstra’s algorithm 6 9 ∞ ∞ 2 7 b c 2 11 ∞ 2 8 3 f 1 a 1 0 5 6 4 e d 5 3 ∞ 5 ∞ S: nodes colored with red Label of vertex: the estimated or determined shortest-path weight Highlighted edges: used in the shortest paths

30. Practice Problem • Find shortest path from f to all other nodes 7 b c 2 2 3 f 1 a 1 5 6 4 e d