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 u=v 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 u=v 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