49 terms

Betsy_McCallTEACHER

Euler circuit problem

A graph problem where one must travel across every edge of the graph just once

Chinese postman problem

A kind of Euler circuit/path problem using real neighborhoods

unicursal tracings

a kind of Euler circuit/path problem using drawings in which one traces all the edges without lifting the pencil

graph

a picture consisting of dots and lines (a structure that defines pairwise relationships within a set of objects)

vertices (singular: vertex)

dots on a graph

edges

lines connecting dots on a graph

loop

an edge connecting a vertex to itself

isolated vertex

a vertex with no edges connecting it

adjacent (vertices)

there is a single edge connecting the vertices

adjacent (edges)

the edges share a single vertex

degree (of a vertex)

the number of edges that connect to that vertex (loops count twice)

odd (vertex)

a vertex with an odd number of edges

even (vertex)

a vertex with an even number of edges

path

a sequence of vertices with the property that each vertex is adjacent to the next one (in an Euler version, every edge must be touched, but need not end in the same place; in a Hamilton version, every vertex must be touched just once but need not end in the same place as it began)

length of path

the number of edges in a path

curcuit

a sequence of vertices with the property that each vertex is adjacent to the next one (in an Euler version, every edge must be touched, and must end at the same vertex it started on; in a Hamilton version, every vertex must be touched just once and ends on the same vertex it started on)

connected (graph)

if, given any two vertices in the the graph, there is a path between those two vertices

bridge

an edge which, if removed from the graph, will disconnect the graph

Euler path

every edge must be touched, but need not end in the same place

Euler circuit

every edge must be touched, and must end at the same vertex it started on

Hamilton path

every vertex must be touched just once but need not end in the same place as it began

Hamilton circuit

every vertex must be touched just once and ends on the same vertex it started on

Euler's Circuit Theorem

a connected graph has an Euler path if every vertex is even

Euler's Path Theorem

a connected graph has an Euler path if there are exactly two odd vertices

Fleury's Algorithm

an algorithm that helps find Euler's paths/curcuits

eulerizing a graph

the process that allows one to duplicate existing edges of a graph with too many odd vertices to create an Euler circuit (eliminates the odd vertices by repeating edges)

semi-eulerizing a graph

the process that allows one to duplicate existing edges of a graph with too many odd vertices to create an Euler path (eliminates all but two odd vertices by repeating edges)

Traveling Saleman Problem (TSP)

a kind of real-world Hamilton circuit problem

complete graph

a graph with N vertices in which every pair of distinct vertices is connected with an edge (and no multiple edges or loops)

K5 complete graph

K16 complete graph

K4 complete graph

weighted graph

a graph where each edge is assigned a weight (such as for mileage, cost, etc.)

Brute Force

a method of finding Hamilton circuits in which every possible weighted circuit is found, analyzed for cost, and the cheapest among the exhaustive list is selected

nearest-neighbor algorithm

a method of finding an approximately best Hamilton circuit by selecting a starting vertex and choosing edges that are the least costly (lowest weight) from the current position without returning to the starting point until all vertices are visited

repeated nearest-neighbor algorithm

repeating the nearest-neighbor algorithm from every vertex and comparing these to find the cheapest among them

optimal algorithm

an algorithm guaranteed to find the cheapest circuit (once complete)

approximate algorithm

an algorithm not guaranteed to find the cheapest, but can find relatively cheaper solutions

efficient algorithm

an algorithm that can be executed with limited computational time and can achieve at least a reasonable proportion of the optimal cost

inefficient algorithm

an algorithm that requires significant computational effort to achieve disproportionately small gains in cost savings

cheapest link algorithm

an algorithm that selects from the cheapest edges in a graph to construct a Hamilton circuit, avoiding visiting any vertex more than 2 times

subgraph

a subset of edges from a larger graph

network

an alternate name for a graph

tree

a network with no circuits

spanning tree

a subgraph of a network that connects all the vertices of a network but has no circuits

minimum spanning tree (MST)

a spanning tree of a weighted network with minimum cost

redundant edge

an edge in a graph that is not a bridge

redundancy

the number of redundant edges in a graph

Kruskal's algorithm

similar algorithm to cheapest link but for finding minimum spanning trees