3. Components
• A component of a graph is a maximal
connected subgraph
Ex.
• A component is a subgraph that cannot be
extended while preserving its connectedness by
including any additional vertices and/or edges
• Every connected graph has exactly one
component and every disconnected graph has
at least two components. 3
4. Cut Vertices
• A cut vertex of graph G is a vertex 𝑣 such that
when the vertex 𝑣 are removed, the number of
components are increased, i.e., 𝑐(𝐺 − 𝑣) >
𝑐(𝐺).
• 𝐺:
• Thus if 𝑣 is a cutpoint/cut vertex of a connected
graph G, then 𝐺 − 𝑣 is disconnected.
4
v
Removal of 𝑣
5. • A cut vertex of a graph is a separating vertex
• However a separating vertex need not to be a
cut vertex. Consider a vertex with a loop and at
least one other edge.
5
Cut Vertex
6. 6
• It may not be a cut vertex (depending on its
neighbours) but it is a separating vertex since
one of the parts of the partition in the definition
of separation is the loop and the separating
vertex.
Cut Vertex
7. Bridge
• A bridge in a graph 𝐺 is an edge 𝑒 𝜖 𝐸(𝐺) such
that 𝑐(𝐺 − 𝑒) > 𝑐(𝐺).
7
* In fact, if 𝒆 is a bridge then
𝒄 𝑮 − 𝒆 → 𝒄 𝑮 + 𝟏
Removal of 𝑒
A graph containing one or more bridges is
said to be a bridged graph.
8. Bridge
• In some books a bridge is sometimes refer to
as isthmus, cut – edge, or cut-arc.
• Likewise, a graph that has no bridge at all is
referred to as a bridgeless graph or isthmus-
free graph.
8
9. Excercises
• Determine all cut vertices and all bridges in the
graph below:
• Cut Vertices = ?
• Bridges = ?
Cut Vertices = 3
Bridges = 2
9
a
b
d
c e
f g
h
j
i
k
l
m
n
o
p
q
r
10. nonseparable
• A nontrivial connected graph is called
nonseparable or two-connected if it has no
cutvertices.
• There are two nonseparable graphs on one
vertex namely 𝐾1 and 𝐾1 with a loop.
10
11. • Definition: A block is a maximal nonseparable
subgraph.
• Ex
• 𝐺:
𝐺:
11
Block
12. Block
12
• 𝐺:
Note that: 𝑐 is a cut point. Thus we only have two
blocks
a
b
d
c
e
f
Note that is not a block
Since its not maximal nonseparable.
14. • Thm. If 𝐺 is a graph with at least one cut vertex,
then at least two of the blocks of 𝐺 contain
exactly one cut vertex. These are called end
blocks.
14
End block
