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# Data structures and algorithms lab8

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• 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.