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