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PROFESSOR: The
point of switching

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from relational language to
graph theoretical language

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is really so that we can talk
about paths and connections.

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So let's look at the topic of
graph connectivity in general.

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Two vertices in a simple
graph, or for that matter,

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a directed graph, are said
to be connected if and only

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if there's a path between them.

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In a directed graph, the
path would have a direction.

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In a simple graph, paths
don't have direction.

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So a is connected to b if and
only if b is connected to a.

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It's a symmetric relation.

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So two vertices are
connected if and only

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if there's a path between them.

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That's equivalent, of
course, to saying if and only

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if there's a walk between them.

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We include length 0
paths and length 0 walks,

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so every vertex is considered
to be connected to itself.

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A whole graph is said to be
connected if all of its vertex

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are connected to each other.

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So every graph you
can think of as

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broken up into the mutually
connected pieces, or subgraphs,

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which are called its
connected components.

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So let's look at
a simple example.

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Let's look at the connections
between MIT buildings,

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where we draw an edge between
building 10 and building 13

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if there's a door between them
or a corridor or between them.

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So there's a corridor between
building 10 and building 4,

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but not between building
10 and building 12.

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To get to 12, you
have to go through 4.

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That's the main building numbers
off the MIT Infinite Corridor.

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East campus, of course,
isn't connected to anything,

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so it's a single,
isolated vertex.

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And then there's the medical
center in E17 and E25,

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which are a sequence of four
buildings that are connected

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as indicated, but not
connected at all to east campus

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or the Infinite
Corridor, that is,

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you have to go outside
to get from east campus

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to the Infinite Corridor or
from the Infinite Corridor

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to the medical center.

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Unless you sneak through the
basement and another restricted

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areas.

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So this is one graph.

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It's not three graphs.

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This is one graph
which has three parts,

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and so it has three
connected components.

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In general, the more
connected components

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a graph has, the
more broken up it

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is, and that's a
way to remember it.

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The formal definition of
the connected component

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of a vertex, v, is simply
all of the vertices, w,

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that are connected
to v. And if you

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look at these
connected components,

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they've defined an equivalence
relation on the vertices,

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of course, because a
connected component is a block

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of the equivalence relation.

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It's a block of the
partition associated

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with the equivalence relation.

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Another way to define this,
the set of w that are connected

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to v, is simply it's
taking the image of v

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under the greater than or
equal to 0 walk relation.

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E star is our notation
for the walk relation

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in the graph whose edges are
E, including walks of length 0.

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So a graph is connected then
means it really has only one

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connected component.