Data structures and algorithms lab8

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This document summarizes key concepts about connected graphs, Hamiltonian graphs, and Eulerian graphs. It provides exercises to check if a graph is connected, Hamiltonian, or Eulerian. It defines a connected graph as one where every vertex is connected by a path. Hamiltonian graphs contain a path visiting every vertex once, and Eulerian graphs contain a cycle visiting every edge once in a connected graph where all vertices have even degree.

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

OBJECTIVES
 Connected Graph
 Hamiltonian Graph
 Eulerian Graph

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.

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.

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.

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

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.

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

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.

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

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

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.

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.

HOMEWORK
Finish all the lab assignments.

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Data structures and algorithms lab8

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.

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