49 terms

MAT 100 Exam #4 Vocabulary Review

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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