Math in the Modern World- Test 4
connected
A graph is connected if there is a path from any vertex to any other vertex.
What is the main difference between the Euler path and Hamilton path?
Edges and Vertices. A Euler path contains all the edges of a graph, while a Hamilton path passes through all the vertices.
directed graph
a graph in which all edges are directed
complete graph (Kn)
a graph in which every pair of vertices is joined by an edge. A complete graph with n vertices is denoted Kn.
Eulerian graph
a graph with all even vertices (no odds) and contains an Euler circuit
Euler path
a path containing all the edges of a graph (means that the graph is traceable)
Hamilton path
a path that passes through all the vertices of a graph exactly once
graph
consists of a finite set of points, called vertices, and lines called edges that join the pairs of vertices.
Hamiltonian
if a graph has a Hamilton circuit
even
if a vertex is the endpoint of an even number of edges (0,2,4,6,8)
odd
if a vertex is the endpoint of an odd number of edges (1,3,5,7,9)
complete
if every pair of vertices in a graph is joined by an edge (i.e. all vertices are directly connected)
directed path
a sequence of edges from one vertex to another, following the indicated directions of the edges within the graph
path
a series of consecutive edges in which no edge is repeated
Euler circuit
an Euler path that begins and ends at the same vertex
Fleury's Algorithm
an algorithm for finding Euler circuits or Euler paths in a graph; it builds the Euler circuit (path) edge by edge- choosing a bridge of the yet-to-be traveled part of the graph only when there is no other choice
Brute Force Algorithm
an algorithm that checks the cost of every possible Hamilton circuit and chooses the optimal one
bridge
an edge in a connected graph without which the graph would be disconnected
Nearest Neighbor Algorithm
starts at a designated vertex and at each step it visits the nearest neighbor (among the vertices not yet visited) until the tour is completed
length
the number of edges in a path
degree
the number of edges joined to that vertex
weights
the numbers on the edges
Eulerize a graph
the process of adding edges to a non-Eulerian graph so that the new graph has only even vertices and an euler circuit can be determined
weight
the sum of the weights of the edges
Hamilton circuit
when a Hamilton path begins and ends at the same vertex
directed edge
when an edge has a direction
weighted
when we assign numbers to the edges of a graph