Use the technique presented in Example. We do not list duplicates in the tables of information.
Scheduling meetings. Repeat Exercise using the following table.
Committee | Has Members in Common With |
Academic standards | Scholarship, university advancement |
Computer use | University advancement, affirmative action |
Campus beautification | Curriculum, academic festival, faculty union, event scheduling, scholarship |
Affirmative action | Scholarship |
University advancement | Curriculum, academic festival, faculty union |
Faculty union | Computer use, event scheduling, academic festival |
Long-range planning | Campus beautification, computer use |
Using Graph Theory to Schedule Committees
Each member of a city council usually serves on several committees to oversee the operation of various aspects of city government. Assume that council members serve on the following committees: police, parks, sanitation, finance, development, streets, fire department, and public relations.
Use Table, which lists committees having common members, to determine a conflict-free schedule for the meetings. We do not duplicate information in Table. That is, because police conflicts with fire department, we do not also list that fire department conflicts with police.
Committee | Has Members in Common With |
Police | Public relations, fire department |
Parks | Streets, development |
Sanitation | Fire department, parks |
Finance | Police, public relations |
Development | Streets |
Streets | Fire department, public relations |
Fire department | Finance |
SOLUTION:
We first will model the information in this table with the graph in Figure. We join two committees by an edge provided they have a conflict. This problem is similar to the map-coloring problem. If we color this graph, then all vertices having the same color represent committees that can meet at the same time. We show one possible coloring of the graph in Figure.
Graph model of committees with common members.
From Figure, we see that the police, streets, and sanitation committees have no common members and therefore can meet at the same time. Public relations, development, and the fire department can meet at a second time. Finance and parks can meet at a third time.
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