Agenda:Problem statement

Graph Construction

Relation to graph problem

Special Property

Problem solution

References

BAND WIDTH ALLOCATION USING GRAPH THEORY

GOVARDHANA SHRAVAN

Problem Definition:A township has to be setup with few radio FM towers in a cost effective way where they have to setup with minimum number of frequency ranges and it has to be taken care that the neighbour intersecting tower should have different frequency. We have to design a problem such that the adjacent towers should not collide with the frequency

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Graph construction:1

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Relation to a graph problem:

In the given problem let us represent the graph to be a chordal graph.

G = (V,E) is a finite undirected graph with vertex set V and edge set E, |V| =

n,

A simple Graph G is triangulated if every cycle of length > 3 has an edge joining 2 nonadjacent vertices of the cycle. The edge is called a chord, and triangulated graphs are also called chordal graphs.

Special property: A perfect vertex elimination scheme of a graph G is an ordering {v1, v2, v3, ..., v n } such that for 1 ≤ i ≤ n-1, vi is a simplicial vertex of the subgraph of G induced by {vi, vi+1, vi+2, ..., v n }. It is also called a perfect scheme.

A perfect elimination ordering in a graph is an ordering of the vertices of the graph such that, for each vertex v, v and the neighbors of v that occur after v in the order form a clique. A graph is chordal if and only if it has a perfect elimination ordering

Problem solution:

For the graph with perfect elimination order, we use coloring property in order to find the solution.The vertices of the graph are colored in such a way that no two adjacent vertices are of same color. The vertex A is colored with green color. So if then the vertices 2,3,4 are adjacent to vertex 1 as well as each other so they are colored blue, red and black respectievely. Then the vertex 5 is colored again with either green or black so that minimum number of colors are used.In our graph 4 colors are used. So minimum number of frequencies required to setup for the township are 4.

Chromatic number = 4

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General solution:

In a paper presented by A. Graf, M. Stumpf, and G. Weibenfels on coloring unit disk graphs, UD graphs are the intersection graphs of equal sized disks in plane. Coloring of UD graph problems arise in the study of channel assignment problem in broadcast networks. It is shown that coloring a UD graph remains NP-complete for any fixed number of colors k ≥ 3. furthermore, a new 3-approximation algorithm for the problem is presented which is based on network flow and matching techniques.

References:

- http://en.wikipedia.org/wiki/Intersection_graph.- http://www.ii.uib.no/~pinar/chordal.pdf- http://

citeseerx.ist.psu.edu/viewdoc/summary;jsessionid=7505EC80077FAF5F24D86ABC31A07C30?doi=10.1.1.142.3742

- On Coloring Unit Disk Graphs-A. Graf, M. Stumpf, and G. Weißenfels2.- http://en.wikipedia.org/wiki/Unit_disk_graph

Queries?