Introduction to Graph Theory

1
Introduction to Graph Theory

• Day 3
• Elementary Concepts of Graph Theory

2
Subgraphs

• Let G be a graph. A graph H is a subgraph of G if
  
• and
  If a graph F is isomorphic to a subgraph of
  H of G, then F is also called a subgraph of G.
• Find all subgraphs of K4, up to isomorphism.

3
u-v walk

• Let u and v be vertices in a graph G. A u-v walk
  in G is a sequence of adjacent vertices in G
  starting with u and ending with v.
• Give two different u-v walks for the graph below.

4
u-v trail/path

• A u-v trail in a graph G is a u-v walk which does
  not repeat any edge.
• Are the u-v walks given on the previous slide u-v
  trails?
• A u-v path is a u-v walk which does not repeat
  any vertex.
• Is a u-v path necessarily a u-v trail?

5
Connected

• Two vertices u and v in G are connected if uv or
  there is a u-v path.
• A graph G is connected if every pair of vertices
  in G are connected. Otherwise, G is disconnected.
• The maximal connected subgraphs of G are called
  the components of G. Notice that a connected
  graph has only one component.
• Give an example of a disconnected graph with
  three components.

6
Circuits and Cycles

• A u-v trail in which uv and which contains at
  least one other vertex is called a circuit.
• A circuit which does not repeat any vertices
  (except for the first and last) is called a
  cycle.
• Give an example of a graph for which every
  circuit is a cycle.

7
Exercises

• Let G be a graph of order 13 with three
  components. Explain why one of the components
  must have 5 vertices.
• Let G be a graph of order p where p is even such
  that G has two complete components. Prove that
  the minimum size possible for G is q(p2-2p)/4.
  If G has this size, what does G look like?

8
Exercises

• Let G be a graph, and let R denote the relation
  is connected to on the set V(G). Show that R is
  an equivalence relation. Determine the
  equivalence classes.
• HW page 43, 27, 28, 39, 40

9
Subtraction

• If e is an edge of the graph G, then G-e is the
  subgraph of G with all the same vertices as G and
  all the edges except for e.
• If v is a vertex of G, then G-v is the subgraph
  of G with all the vertices of G except for v and
  all the edges of G except for the edges which are
  incident to v.

10
Cut-vertex

• A vertex v in a connected graph G is a cut-vertex
  if G-v is disconnected.
• Draw a graph with no cut-vertices.
• Draw a graph with two cut-vertices.

11
Bridge

• An edge e in a connected graph G is a bridge if
  G-e is disconnected.
• Draw an example of a graph G with a bridge.
• If e is a bridge in a graph G, how many
  components does G-e have?

12
Bridges

• Theorem 2.5 Let G be a connected graph. An edge
  e of G is a bridge if and only if e does not lie
  on any cycle of G.

13
Exercises

• Let G be a connected graph containing only even
  vertices. Prove that G cannot contain a bridge.
• Let G be a connected graph, and let u, v, and w
  be three vertices of G. Suppose that every u-w
  path contains v. What property does v have? Why?

14
Exercises

• Prove or give counterexample If G is a connected
  graph with a cut-vertex, then G has a bridge.
• HW pages 47-48, 45, 46, 50, 54, 55