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22C:19 Discrete Math Graphs

22C:19 Discrete Math Graphs. Spring 2014 Sukumar Ghosh. Seven Bridges of K ⍥ nigsberg. Is it possible to walk along a route that cross each bridge exactly once?. Seven Bridges of K ⍥ nigsberg. A Graph. What is a Graph. A graph G = (V, E) consists of V , a nonempty set of vertices

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22C:19 Discrete Math Graphs

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  1. 22C:19 Discrete MathGraphs Spring 2014 Sukumar Ghosh

  2. Seven Bridges of K⍥nigsberg Is it possible to walk along a route that cross each bridge exactly once?

  3. Seven Bridges of K⍥nigsberg

  4. A Graph

  5. What is a Graph A graphG = (V, E) consists of V, a nonempty set of vertices (or nodes) and E, a set of edges. Each edge connects a pair of nodes that are called its endpoints. Graphs are widely used to model various systems in the real world

  6. Back to the Bridges of K⍥nigsberg

  7. Euler’s solution

  8. Euler path

  9. Simple graph

  10. Types of graph

  11. Definitions At most one edge between a pair of nodes Multiple edges between some pair of nodes Simple graphs vs. multi-graphs Undirected vs. directed graphs (digraphs) Each edge between a pair (u, v) of nodes is directed, and represents an ordered pair

  12. More examples of graphs Each node denotes an actor or an actress, and each edge between P and Q denotes that P, Q worked together in some movie. It is an undirected graph Hollywood graph Web graphs Each node denotes a web page, and each edge from page P to Q Q denotes a link on page P pointing to page Q. It is a directed graph

  13. Application: Exam scheduling

  14. Problems in a computer network

  15. Application: graph orientation

  16. Vertex degree

  17. Degree sequence

  18. Handshaking theorem

  19. Handshaking theorem

  20. A theorem THEOREM. An undirected graph has even number of vertices of odd degree. Can you prove this? It should follow from the handshaking theorem.

  21. Review of basic definitions

  22. Review of basic definitions

  23. Types of graphs

  24. Types of graphs The n-cube graph n=3 Complete graph: All vertices are adjacent Wheel graph

  25. Types of graphs Bipartite graph A simple graph is called bipartite if its vertex set V can be partitioned into two disjoint subsets V1 and V2, such that every edge in the graph connects a vertex in V1 to a vertex in V2. Can always be colored using two colors.

  26. Subgraphs

  27. Computer representation of graphs ADJACENCY MATRIX (taken from Wolfram Mathworld)

  28. Computer representation of graphs ADJACENCY LIST 1 2 4 3 Can be represented as a linked list

  29. Graph isomorphism Taken from MIT 6.042J/18.062J

  30. Graph isomorphism Taken from MIT 6.042J/18.062J

  31. Graph isomorphism Taken from MIT 6.042J/18.062J

  32. Connectivity • An undirected graph is connected if there is a path between • every pair of distinct vertices of the graph. • A connected componentis the maximal connected subgraph • of the given graph.

  33. Connectivity • Erdös number in academic collaboration graph • Erdös number = n means the person collaborated • with someone whose Erdös number is (n-1) • Kevin Bacon number in Hollywood graph • Actor Kevin Bacon once remarked that he worked with • everybody in Hollywood, or someone who worked with them

  34. Cut vertex, cut set, cut edge A cut vertex (or articulation point ) is a vertex, by removing which one can partition the graph. If multiple vertices need to be remove to partition the graph, then the minimal set of such edges a cut set.

  35. Connectivity in directed graphs A directed graph is stronglyconnected if there is a path from any vertex a to any other vertex b of the graph. A directed graph is weaklyconnected if there is a path between any two vertices of the underlying undirected graph.

  36. More definitions Vertex cover is a famous problem in graph theory

  37. Vertex Cover A vertex-cover of an undirected graph G=(V,E) is a subset V’ of V such that if edge (u, v) is an edge of G, then u is in V, v is in V, or both. The set V′ is said to "cover" the edges of G Minimal or minimum vertex cover is a famous problem in graph theory

  38. Dominating Set A dominating setfor a graphG = (V, E) is a subsetD of V such that every vertex not in D is adjacent to at least one member of D. (from Wikipedia) Computing a minimal or minimum dominating set is a famous problem in graph theory

  39. Independent Set Given a graphG = (V, E) an independent set is a subset of vertices no two of which are adjacent to one another. (from Wikipedia) Computing the maximal independent set is a well-known problem in graph theory

  40. Euler path vs. Hamiltonian path Hamiltonian path = A path that passes through every vertex exactly once. A closed path is a Hamiltonian circuit or cycle. Euler path = A path that includes every edge exactly once. A closed path is a Euler circuit or cycle. We have reviewed Euler path in the 7-bridges of Konigsberg Problem.

  41. Hamiltonian path 1 2 4 5 3 Does the above graph have a Hamiltonian cycle? No! Hamiltonian circuit/cycle colored red

  42. Shortest path Weighted graph. Compute the shortest path from a to z

  43. Shortest path: Dijkstra’s algorithm Read the algorithm from page 712 of your text book

  44. Shortest path: Dijkstra’s algorithm

  45. Shortest path: Dijkstra’s algorithm

  46. Traveling Salesman Problem (TSP) A traveling salesman wants to visit each of n cities exactly once, and then return to the starting point. In which order should he visit the cities to travel the minimum total distance? TSP = Computing the minimum cost Hamiltonian circuit. TSP is an extremely hard problem to solve (NP-complete) An optimal TSP tour through Germany’s largest cities (Source: Wikipedia)

  47. Planar Graph A planar graph is one that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges do not intersect except only at their endpoints. planar K4 Butterfly planar Non-planar Non-planar K5 K3,3

  48. Planar Graph How to verify that a graph is a planar graph? It should not depend upon how you draw the graph. Theorem. Any graph that contains a subgraph K5 or K3,3 is not planar. Theorem. Any planar graph can be colored using at most four colors. {This is known as the four-color theorem or the four-color conjecture}

  49. Graph Coloring Let G be a graph, and C be a set of colors. Graph coloring finds an assignment of colors to the different nodes of G, so that no two adjacent nodes have the same color. The problem becomes challenging when the |C| is small. Chromatic number. The smallest number of colors needs to color a graph is called its chromatic number.

  50. Graph Coloring What are the chromatic numbers of these two graphs?

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