Connected Components
Mathematics for Computer Science
MIT 6.042J/18.062J
Every graph consists of
separate connected
pieces (subgraphs) called
Graph Connectivity
Trees
Albert R Meyer, March 12, 2010
connected components
lec 6F.1
Connected Components
13
12
26
10
4
8
16 66
lec 6F.2
Connected Components
E17
East Campus
Albert R Meyer, March 12, 2010
E25
Med Center
The connected component
of vertex v ::=
Infinite corridor
3 connected components
the more connected components,
the more “broken up" the graph is.
Albert R Meyer, March 12, 2010
lec 6F.3
lec 6F.4
Edge Connectedness
Connected Components
Def: vertices v, w are
k-edge connected
if they remain connected
whenever fewer than k
edges are deleted.
So a graph is connected
iff it has only
1 connected component
Albert R Meyer, March 12, 2010
Albert R Meyer, March 12, 2010
lec 6F.5
Albert R Meyer, March 12, 2010
lec 6F.6
1
k-edge Connectedness
Edge Connectedness
no path
no path
delete
1-edge connected
Albert R Meyer, March 12, 2010
2-edge connected
lec 6F.8
Edge Connectedness
Albert R Meyer, March 12, 2010
lec 6F.9
k-edge Connectedness
no path
Def: A whole graph is k-
edge connected iff every
two vertices are k-edge
connected.
3-edge connected
Albert R Meyer, March 12, 2010
lec 6F.10
Edge Connectedness
lec 6F.11
k-edge Connectedness
this whole graph is
Connectivity measures fault
tolerance of a network:
how many connections can
fail without cutting off
communication?
Albert R Meyer, March 12, 2010
Albert R Meyer, March 12, 2010
delete
1-edge connected
lec 6F.12
Albert R Meyer, March 12, 2010
lec 6F.13
2
Cut Edges
Cut Edges
An edge is a cut edge if
removing it from the graph
disconnects two vertices.
Albert R Meyer, March 12, 2010
B
B is a cut edge
lec 6F.16
Cut Edges
Albert R Meyer, March 12, 2010
lec 6F.18
Cut Edges
A
deleting B gives
two components
Albert R Meyer, March 12, 2010
A is not a cut edge
lec 6F.19
Cut Edges
lec 6F.20
Cut Edges
So a connected graph is
2-edge connected iff
it has no cut edge.
still connected with
edge A deleted
Albert R Meyer, March 12, 2010
Albert R Meyer, March 12, 2010
lec 6F.21
Albert R Meyer, March 12, 2010
lec 6F.22
3
Cycles
Cycles
A cycle is a path that begins
and ends with same vertex
A cycle is a path that begins
and ends with same vertex
a
v
a
v
w
b
w
b
path: v···b···w···w···a···v
also: a ···w ···w ···b ···v ···a
also: a···v···b···w···w···a
lec 6F.23
Albert R Meyer, March 12, 2010
Simple Cycles
A simple cycle is a cycle
of length > 2 that doesn’t
cross itself:
v
w
a
path: v···a···w···v
lec 6F.25
Simple Cycles
also: w···a···v···w
Albert R Meyer, March 12, 2010
lec 6F.26
Cut Edges and Cycles
length > 2 implies that
going back & forth over
an edge is not a simple cycle
w
Albert R Meyer, March 12, 2010
w
a
path: v···a···w···v
Albert R Meyer, March 12, 2010
v
lec 6F.24
Simple Cycles
A simple cycle is a cycle
of length > 2 that doesn’t
cross itself:
v
Albert R Meyer, March 12, 2010
lec 6F.27
Lemma: An edge is a
not a cut edge iff
it is on a simple cycle.
Albert R Meyer, March 12, 2010
lec 6F.30
4
Trees
Trees
A tree is a connected graph
with no simple cycles.
A tree is a connected graph
with every edge a cut edge.
equivalently:
Albert R Meyer, March 12, 2010
lec 6F.31
More Trees
equivalently
Albert R Meyer, March 12, 2010
lec 6F.32
Other Tree Definitions
• graph with a unique simple
path between any 2 vertices
• connected graph with n
vertices and n–1 edges
• an edge-maximal acyclic graph
Albert R Meyer, March 12, 2010
lec 6F.33
Albert R Meyer, March 12, 2010
lec 6F.34
Team Problems
Problems
1&2
Albert R Meyer, March 12, 2010
lec 6F.40
5
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6.042J / 18.062J Mathematics for Computer Science
Spring 2010
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