More About Graphs More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Pa[.]
More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan Chapter More About Graphs Discrete Structures for Computing on October 27, 2015 Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring Huynh Tuong Nguyen, Tran Vinh Tan Faculty of Computer Science and Engineering University of Technology - VNUHCM 9.1 Acknowledgement More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan Contents Connectivity Some slides about Euler and Hamilton circuits are created by Chung Ki-hong and Hur Joon-seok from KAIST Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring 9.2 Contents More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Contents Connectivity Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Planar Graphs Graph Coloring Graph Coloring 9.3 More About Graphs Paths and Circuits a b c Huynh Tuong Nguyen, Tran Vinh Tan Contents d e f Connectivity Paths and Circuits Euler and Hamilton Paths Simple path of length Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem a b c Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring d e f Circuit of length 9.4 More About Graphs Path and Circuits Huynh Tuong Nguyen, Tran Vinh Tan Definition (in undirected graph) • Path (đường đi) of length n from u to v: a sequence of n edges {x0 , x1 }, {x1 , x2 }, , {xn−1 , xn }, where x0 = u and xn = v • A path is a circuit (chu trình) if it begins and ends at the same vertex, u = v Contents Connectivity Paths and Circuits • A path or circuit is simple (đơn) if it does not contain the same edge more than once Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem a b c a b c Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring d e Simple path f d e f Not simple path 9.5 Path and Circuits More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan Contents Connectivity Definition (in directed graphs) Path is a sequence of (x0 , x1 ), (x1 , x2 ), , (xn−1 , xn ), where x0 = u and xn = v Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring 9.6 More About Graphs Connectedness in Undirected Graphs Huynh Tuong Nguyen, Tran Vinh Tan Definition • An undirected graph is called connected (liên thông ) if there is a path between every pair of distinct vertices of the graph • There is a simple path between every pair of distinct vertices of a connected undirected graph Contents Connectivity Paths and Circuits d e b Euler and Hamilton Paths f Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem a c h g Planar Graphs Graph Coloring Connected graph Disconnected graph Connected components (thành phần liên thông ) 9.7 More About Graphs How Connected is a Graph? Huynh Tuong Nguyen, Tran Vinh Tan d a f g Contents Connectivity Paths and Circuits c bb e Euler and Hamilton Paths h Euler Paths and Circuits Hamilton Paths and Circuits Definition Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm • b is a cut vertex (đỉnh cắt) or articulation point (điểm khớp) What else? • {a, b} is a cut edge (cạnh cắt) or bridge (cầu) What else? Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring 9.8 More About Graphs How Connected is a Graph? b d Huynh Tuong Nguyen, Tran Vinh Tan g Contents a e Connectivity Paths and Circuits Euler and Hamilton Paths Definition • This graph don’t have cut vertices: nonseparable graph (đồ thị phân tách) Euler Paths and Circuits Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm • The vertex cut is {c, f }, so the minimum number of vertices in a vertex cut, vertex connectivity (liên thông đỉnh) κ(G) = Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring • The edge cut is {{b, c}, {a, f }, {f, g}}, the minimum number of edges in an edge cut, edge connectivity (liên thông cạnh) λ(G) = 9.9 Applications of Vertex and Edge Connectivity More About Graphs Huynh Tuong Nguyen, Tran Vinh Tan • Reliability of networks • Minimum number of routers that disconnect the network • Minimum number of fiber optic links that can be down to disconnect the network Contents Connectivity Paths and Circuits Euler and Hamilton Paths Euler Paths and Circuits • Highway network • Minimum number of intersections that can be closed • Minimum number of roads that can be closed Hamilton Paths and Circuits Shortest Path Problem Dijkstra’s Algorithm Bellman-Ford Algorithm Floyd-Warshall Algorithm Ford’s algorithm Traveling Salesman Problem Planar Graphs Graph Coloring 9.10