1. Chapter 8 Topics in Graph Theory
CSCI 115
Chapter 8
Topics in Graph Theory

2. CSCI 115
§8.1
Graphs

3. §8.1 – Graphs
Graph
A graph G consists of a finite set of vertices V, a finite set of edges E, and a function γ that assigns a subset of vertices {v, w} to each edge (v may equal w)
If e is an edge, and γ(e) = {v, w} we say e is the edge between v & w, and that v & w are the endpoints of e

4. §8.1 – Graphs Terminology Degree of a vertex Loop Isolated vertex
Number of edges having that vertex as an endpoint
Loop
Edge from a vertex to itself
Contributes 2 to the degree of a vertex
Isolated vertex
Vertex with degree 0
Adjacent vertices
Vertices that share an edge

5. §8.1 – Graphs
Path
A path  in a graph G consists of a pair of sequences (V, E), V: v1, v2, …, vk and E: e1, e2, …, ek–1 s.t.:
γ(ei) = {vi, vi+1} i
ei  ej  i, j

6. §8.1 – Graphs More terminology Circuit Simple path Simple circuit
Path that begins and ends at the same vertex
Simple path
No vertex appears more than once (except possibly the first and last)
Simple circuit
Simple path where first and last vertices are equal

7. §8.1 – Graphs Special types of graphs Connected graph
 path from every vertex to every other (different) vertex
Disconnected graph
There are at least 2 vertices which do not have a path between them
components
Regular Graph
All vertices have the same degree

8. §8.1 – Graphs Special families of graphs (n  Z+)
Un: The discrete graph on n vertices
The graph with n vertices and no edges
Kn: The complete graph on n vertices
The graph with n vertices, and an edge between every pair of vertices
Ln: The linear graph on n vertices
The graph with n vertices and edges {vi, vi+1}  i  {1, 2, …, n – 1}

9. §8.1 – Graphs
Subgraphs
If G = (V, E, γ) and E1  E, V1  V s.t. V1 contains (at least) all of the end points of edges in E1, then H = (V1, E1, γ) is a subgraph of G
If G = (V, E, γ) and e  E, then Ge is the subgraph found by deleting e from G and keeping all vertices

10. §8.1 – Graphs Quotient graphs
If G = (V, E, γ) and R is an equivalence relation on V, then GR is the quotient graph found by merging all vertices within the same equivalence classes.

11. §8.2 Euler Paths and Circuits
CSCI 115
§8.2
Euler Paths and Circuits

12. §8.2 – Euler Paths and Circuits
A path in a graph G is an Euler path if it includes every edge exactly once
An Euler circuit is an Euler path that is also a circuit

13. §8.2 – Euler Paths and Circuits
Theorem 8.2.1
If G has a vertex of odd degree, there can be no Euler circuit of G
If G is a connected graph and every vertex has even degree then there is an Euler circuit in G
Theorem 8.2.2
If a graph G has more than 2 vertices of odd degree, there can be no Euler path in G
If G is connected and has exactly 2 vertices of odd degree, then there exists an Euler path in G. Any Euler path must begin at one vertex of odd degree, and end at the other.

14. §8.2 – Euler Paths and Circuits
Theorem and 8.2.2
Existence theorems
Bridge
A bridge is an edge in a connected graph that if removed would result in a disconnected graph

15. §8.2 – Euler Paths and Circuits
Fleury’s Algorithm for finding an Euler circuit for a connected graph where every vertex has even degree (let G = V, E, γ)
Select and edge e1 of G with vertices (v1, v2) that is not a bridge. Let  be (V: v1, v2, E: e1). Remove e1 from E and v1 and v2 from V to create G1.
Suppose V: v1, v2, …, vk and E: e1, e2, …, ek–1 have been constructed to form Gk–1. Select ek in Gk–1 that has vk as a vertex and is not a bridge in Gk–1 (discounting vk). Extend V to v1, v2, …, vk, vk+1 and E to e1, e2, …, ek–1, ek.
Repeat step 2 until no edges remain in E.

16. §8.3 Hamiltonian Paths and Circuits
CSCI 115
§8.3
Hamiltonian Paths and Circuits

17. §8.3 – Hamiltonian Paths and Circuits
A Hamiltonian path is a path that contains each vertex exactly once
A Hamiltonian circuit is a Hamiltonian path that is also a circuit

18. §8.3 – Hamiltonian Paths and Circuits
Theorem 8.3.1
Let G be a connected graph with n vertices (n  Z, n  2), with no loops or multiple edges. G has a Hamiltonian circuit if for any 2 vertices u and v of G that are not adjacent, the degree of u plus the degree of v is  n.
Corollary 8.3.1
G has a Hamiltonian circuit if each vertex has degree  (n/2).

19. §8.3 – Hamiltonian Paths and Circuits
Theorem 8.3.2
Let the number of edges of G be m. Then G has a Hamiltonian circuit if m  ½(n2 – 3n + 6) where n is the number of vertices.