Lecture 19: CONNECTIVITY Sections 8.1 8.3

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Lecture 19 CONNECTIVITYSections 8.1 - 8.3
CS1050 Understanding and Constructing Proofs
Spring 2006

• Jarek Rossignac

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Lecture Objectives

• Learn graph terminology

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What are the types of graph?

• Graph G(V,E)
• V set of vertices (non-empty)
• E set of edges (unordered pairs of distinct
  elements of V)

Loop
Multiple edge
Simple Graph
Multigraph
Directed graph
Pseudograph
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Graph Types
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Examples of graphs

• Simple
• Multigraph (multiple edges)
• Pseudograph (multiple edges and loops)
• Directed (loops)
• Directed multigraph (multiple edges and loops)

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Classify graphs

• Edge between A and B means
• They know each other
• A is a parent of B
• They compete
• A has called B
• Page A has a link to page B
• Have collaborated
• A has beaten B in round-robin

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What is adjacency and incidence?

• In an undirected graph
• An edge E between vertices A and B is incident
  with them.
• A and B are the endpoints of E
• E connects A and B
• Vertices A and B are adjacent (neighbors) when
  there is an edge incident with both

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What is the degree of a vertex?

• In an undirected graph with e edges
• The degree deg(V), also called valence, of vertex
  V is the number of times V is used by an edge
  (twice by an incident loop).
• A vertex with degree one is pendent (dead end).
• A vertex with degree zero is isolated.
• The sum of the degrees of all vertices if 2e.
• There is an even number of edges of odd degree.

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Directed graph terminology

• E is a directed edge from A to B (denoted A?B)
• A is adjacent to B
• A is the initial vertex of E
• B is adjacent from A
• B is the terminal or end vertex of E
• AB if E is a loop
• In-degree deg(V) of vertex V is the number of
  edges for which it is a terminal vertex
• Out-degree deg(V) of vertex V is the number of
  edges for which it is an initial vertex

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Cycles

• A cycle Cn is has n vertices and n-edge the form
  a cycle
• C3 is a triangle

C5
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Complete graphs Kn

• A complete graph Kn of n vertices is a simple
  graph with one edge between each pair
• K3 is a triangle

K5
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Wheels

• A wheel Wn is a cycle with n vertices plus an
  additional vertex connected to all

W5
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Bipartite graphs

• A graph is bipartite when itd vertices can be
  colored (red/green) so that each edge joins
  vertices of different colors
• It is complete bipartite if there is an edge
  between each pair of vertices of different color

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Subgraph

• A subgraph of G has a subset of the edges and
  vertices of G
• It must include all the vertices bounding all its
  edges!

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Representing graphs

• Vertices (x,y) , edges (a,b)
• Adjacency list vertices (x, y, a, b, )
• Adjacency matrix
• Simple graphs (binary, symmetric)
• Multiple graph integer entries count number of
  edges
• Loops on diagonal
• Incidence matrix edges/vertices
• Two 1s per column

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Isomorphism

• Two graphs G and H are isomorphic if there is a
  bijection between their vertices that leads to
  the same set of edges.
• Expensive to compute, since there are n!
  vertex/label assignments
• Necessary conditions (invariants) help quickly
  decide that two graphs are NOT isomorphic
• same number of vertices and edges
• same degree list

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Assigned Reading

• 8.1, 8.2, 8.3

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Assigned Homework

• P 544-545 3, 4, 5, 6, 7
• P 555 12, 27, 29f, 36, 42
• P 562 1, 10, 38, 39, 49, 57a, 68

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Assigned Project

• P9 Spanning tree