1. DATA STRUCTURES AND ALGORITHMS
LAB 8
Bianca Tesila
FILS, April 2014

2. OBJECTIVES
 Connected Graph
 Hamiltonian Graph
 Eulerian Graph

3. CONNECTED GRAPH
 An undirected graph G = (V, A) is said to be
connected if, for any vertices u and v of S, there is
a chain from u to v.

4. CONNECTED GRAPH – CONNECTED
COMPONENTS
 A connected component (or just component) of
an undirected graph is a subgraph in which any
two vertices are connected to each other by paths,
and which is connected to no additional vertices
in the supergraph.

5. CONNECTED GRAPH
 Classic problems:
 Check whether a graph is connected or not
 Find all the connected components of a graph
!! DFS is used to determine if a graph is connected
or not.

6. CONNECTED GRAPH
!! Exercise:
Check whether a graph is connected or not.

7. HAMILTONIAN GRAPH
 Hamiltonian Path (or traceable path): a path in an
undirected or directed graph that visits
each vertex exactly once.
 Hamiltonian Cycle (or Hamiltonian circuit): a
Hamiltonian path that is a cycle. That
 is, it begins and ends on the same vertex.
 Hamiltonian Graph: a graph that contains a
Hamiltonian cycle
!! Any Hamiltonian cycle can be converted to a
Hamiltonian path by removing one of its edges.

8. HAMILTONIAN GRAPH
https://sites.google.com/site/poggiolidiscretemath/graph-
theory/hamiltonian-graphs

9. HAMILTONIAN GRAPH
!! Exercise:
Check whether a graph is Hamiltonian or not.
Hint:
If G is a non-oriented with n> = 3 vertices, such
that every vertex of G has the degree greater or
equal to N / 2, then G is Hamiltonian graph.

10. EULERIAN GRAPH
 Eulerian Trail (or Eulerian path): a trail in a
graph which visits every edge exactly once. It
can end on a vertex different from the one on
which it began.
 Eulerian Cycle: an Eulerian trail which starts
and ends on the same vertex.
 Eulerian Graph: a graph that contains an
Eulerian cycle

11. EULERIAN GRAPH
 A graph G, without isolated vertices, is Eulerian
if and only if it is connected and the degrees of all
vertices are even numbers.
 Classic problem: find the Eulerian cycle in a
graph.
https://sites.google.com/site/poggiolidiscretemath/gr
aph-theory/eulerian-graphs

12. EULERIAN GRAPH
!! Exercise:
Check whether a graph is Eulerian or not.
Hint:
A connected graph G is an Euler graph if and only
if all vertices of G are of even degree.

13. EULERIAN VS. HAMILTONIAN GRAPHS
!! An Eulerian circuit traverses every edge in a
graph exactly once, but may repeat vertices, while
a Hamiltonian circuit visits each vertex in a
graph exactly once but may repeat edges.

14. HOMEWORK
Finish all the lab assignments.