Download MAT110 Graph Theory Vocabulary

MAT 110 Vocabulary Ch 4
4.1
GRAPH: consists of a finite set of points, called VERTICES, and lines, called EDGES, that join pairs of
vertices.
TRACE a graph: to begin at some vertex and draw the entire graph without lifting your pencil and
without going over any edge more than once.
CONNECTED graph: if it is possible to travel from any vertex to any other vertex of the graph by moving
along successive edges.
BRIDGE: in a connected graph, it is an edge such that if it were removed from the graph then the graph
is no longer connected.
ODD vertex: an endpoint of an odd number of edges
EVEN vertex: an endpoint of an even number of edges
EULER’S THEOREM: a graph can be traced if it is connected and has zero or two odd vertices.
PATH: a series of consecutive edges in which no edge is repeated.
LENGTH: the number of edges in a path
EULER PATH: a path containing all of the edges
EULER CIRCUIT: an Euler path that begins and ends at the same vertex
EULERIAN GRAPH: a graph with all even vertices which contains an Euler circuit
ALGORITHM: a series of steps that we follow to accomplish something
FLEURY’S ALGORITHM: a systematic technique for finding Euler circuits.
1. Begin at any vertex
2. After you have traveled over an edge erase it. If all edges at a particular vertex have been
erased, then erase the vertex also.
3. Travel over an edge that is a bridge only if there is no alternative.
EULERIZING a graph: adding edges to a graph with odd vertices so that the graph is Eulerian. Edges can
only be added by creating duplicate edges between vertices that are already connected by an edge.
FOUR-COLOR PROBLEM: There is no particular procedure, but set up the countries and vertices and the
edges represent a common border between two countries.
MAT 110 Vocabulary Ch 4
4.2
HAMILTON PATH: a path that passes through all the vertices exactly once
HAMILTON CIRCUIT: a Hamilton path that begins and ends at the same vertex. A graph with a Hamilton
circuit is called HAMILTONIAN
Note: in a Hamilton path you do NOT have to trace every edge as with an Euler path
COMPLETE GRAPH: every pair of vertices is joined by an edge----Notation – Kn is a complete graph with
n vertices.
# of Hamilton Circuits in Kn : Kn has (n – 1)! Hamilton circuits
WEIGHTED GRAPH: when numbers are assigned to edges
WEIGHT of a PATH: add together the weights of the edges
BRUTE FORCE Algorithm: 1. List all possible Hamilton circuits, 2. Find the weight of each in step 1, 3.
The circuit with the smallest weight is the solution
NEAREST NEIGHBOR ALGORITHM: 1. Start at any vertex, 2. Choose your path by choosing the edge
with the least weight each time, 3. After all vertices have been chosen, close the circuit by returning to
the starting vertex.
BEST EDGE ALGORITHM: 1. Choose any edge with the smallest weight, 2. Choose any reaming edge
with the smallest weight, 3. Keep repeating step 2 but do not allow a circuit to form until all vertices
have been used, Also because a Hamilton circuit cannot have 3 edges joined to the same vertex, never
allow this to happen.
4.3
DIRECTED EDGE: an edge with a direction
DIRECTED GRAPH: a graph in which all edges are directed
DIRECT or 1 STAGE INFLUENCE: there is a directed path of length 1 from X to Y
2 STAGE INFLUENCE: there is a directed path of length 2 from X to Y
TOTAL INFLUENCE: total number of 1 and 2 stage influences
4.4
PERT Diagram: Each vertex contains the number of months needed for a task. Directed edges show
tasks that must be completed prior to other tasks
CRITICAL PATH: the path requiring the most time from “Begin” to the task.