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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.
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.
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