Graphs

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Graphs
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What good is a graph.

• Maps
• Hypertexts
• Circuits
• Networks
• Matching
• Transactions
• Schedules
• Program structure

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Graphs

• A graph is a set of Vertices and a set of Edges
  that connect pairs of distinct vertices (with at
  most 1 edge connecting any pair of vertices).
• This is a simple graph
• What does this mean??

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Graphs

• Simple graphs have a property about the maximum
  number of edges
• A graph with V vertices has at most
• V(V-1) / 2 edges

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Graphs Terms

• Vertex Node
• Arc Edge Link
• Adjacent - Two graph edges are adjacent if they
  have a vertex in common. Likewise, two vertices
  are adjacent if they are joined by a graph edge.
• Incident - An edge is incident to a vertex if the
  vertex is at one end of the edge. A vertex is
  incident to an edge if that edge is incident to
  the vertex.

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Graphs Terms

• Degree - The number of edges incident to a given
  vertex. Each vertex in a directed graph has both
  an indegree, the number of edges coming in to the
  vertex as well as an outdegree, the number of
  edges going out of the vertex.

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Graphs Terms

• Subgraph - A graph G1 is called a subgraph of
  graph G2, if every vertex of the first graph is a
  vertex in the second graph, every edge of the
  first graph is an edge in the second graph, and
  every edge of the first graph joins the same
  vertices as it does in the second graph. The
  purple vertices with bold edges G2 is a subgraph
  of the larger graph G1.

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Graphs Terms

• Isomorphic The simple graphs G1 and G2 are
  isomorphic if there is one-to-one correspondence
  between their sets of vertices which preserves
  vertex adjacency. For directed graphs the
  one-to-one correspondence must preserve adjacency
  while also preserving relative origins and ends
  of edges.
• Which are isomorphic to each other??

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Graphs Terms

• Path A path in a graph is a set of vertices in
  which each successive vertex (after the first) is
  adjacent to its predecessor in the path.

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Graphs Terms

• Simple Path vertices and edges are distinct
• Cycle Simple path in which the first and last
  vertices are the same.

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Graphs Terms

• Connected Graph A graph is connected if there
  is a path from every vertex to every other vertex
  in the graph.

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Graphs Terms

• Complete Graph A graph where all edges are
  present.

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Graphs Terms

• Spanning Tree
• black edges part of the spanning tree
• pink edges not

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Graphs Terms

• Compliment

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Graphs Terms

• Clique A maximum complete subgraph, or simply
  subgraph, which is not a part of any larger
  complete subgraph. The graph shown below has
  three cliques induced by the sets of vertices 1,
  2, 1, 3, 5, and 3, 4, 5, 6.

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Graphs Terms

• Bipartite A bipartite graph is a simple graph
  where the set of vertices can be separated into
  two subsets, called parts, so that every edge in
  the graph joins vertices from separate parts. For
  example, below shows a simple graph which is also
  a bipartite graph because it may be divided into
  two parts, given by the subsets 1, 2 and 3, 4,
  5, where every edge in the graph goes from a
  vertex in one part to a vertex in the other part.
  

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Matrix Implementation
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List Implementation
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List vs. Matrix

• List Implementation
• Space E V
• Finding an edge is proportion to V
• Sparse Graphs??
• Matrix Implementation
• Space V2
• Finding an edge is constant
• Dense Graphs??

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Graph Traversals

• Breadth First Search
• Depth First Search
• Euler Path