Graph Theory

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Graph Theory

• Graph theory is the study of the properties of
  graph structures. It provides us with a language
  with which to talk about graphs.

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Degree

• The degree of a vertex is the number of edges
  incident upon it.
• The sum of the vertex degrees in any undirected
  graph is even (twice the number of edges).
• Every graph contains an even number of odd-degree
  vertices.

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Connectivity

• A graph is connected if there is an undirected
  path between every pair of vertices.
• The existence of a spanning tree is sufficient to
  prove connectivity.
• The vertex (edge) connectivity is the smallest
  number of vertices (edges) which must be deleted
  to disconnect the graph.

4
Some terms

• articulation vertex ?? biconnected
• bridge ?? edge-biconnected
• Testing for articulation vertices or bridges is
  easy via brute force.

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Cycles

• Eulerian cycle (path) a tour which visits every
  edge of the graph exactly once.
• Actually, it is a circuit, not a cycle, because
  it may visit vertices more than once.
• A mailmans route is ideally an Eulerian cycle,
  so he can visit every street (edge) in the
  neighborhood once before returning home.

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• An undirected graph contains an Eulerian cycle if
  it is connected and every vertex is of even
  degree.
• A Hamiltonian cycle is a tour which visits every
  vertex of the graph exactly once.
• The traveling salesman problem asks for the
  shortest such tour on a weighted graph.

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Planer Graph

• Eulers formulan-mf2
• n of vertices
• m of edges
• f of faces
• Trees mn-1, f1
• Cubes n8, m12, f6

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MST

• Kruskals algorithmstarting from a minimal
  edge
• Prims algorithmstarting from a given vertex
• How about maximum spanning tree
• and Minimum Product spanning tree

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Kruskals Algorithm

• Algorithm Kruskal(G)
• InputG(V, E)??????(undirected weighted
  graph),??Vv0,,vn-1
• OutputG??????(minimum spanning tree, MST)
• T?? //T?MST,????????
• while T????n-1?? do
• ???(u, v),??(u, v)?E,?(u, v)???(weight)??
• E?E-(u, v)
• if ( (u, v)??T?????(cycle) ) then ?(u, v)??
• else T?T?(u, v)
• return T

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Kruskals Algorithm -Construct MST
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Prims algorithm

• Algorithm Prim(G)
• InputG(V, E)??????(undirected weighted
  graph),??Vv0,,vn-1
• OutputG??????(minimum spanning tree, MST)
• T?? //T?MST,????????
• X?vx //????????vx????X?
• while T????n-1?? do
• ??(u, v)?E,??u?X?v?V-X,?(u, v)???(weight)??
• T?T?(u, v)
• X?X?v
• return T

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One-to-all shorted path

• Dijkstra??? Dijkstra???????????(source)?????(dest
  ination)??????(one-to-all)???????

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All-pair shortest path

• Floyd-Warshall??????????(all-pair shortest
  path)???