GRAPH COLORING AND CLASSIFYING TROPICAL FISH By Vennam Chandrasekhar Reddy.
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Transcript of GRAPH COLORING AND CLASSIFYING TROPICAL FISH By Vennam Chandrasekhar Reddy.
GRAPH COLORING AND CLASSIFYING TROPICAL
FISH
By
Vennam Chandrasekhar Reddy
AGENDA
Problem statement
Graph Construction
Relation to graph problem
Special Property
Problem solution
Comments
References
PROBLEM DEFINITION:
A tropical fish hobbist had six different types of fish: Alphas, Betas, Certas, Deltas, Epsalas and Fetas which are designated by A, B, C, D, E, and F respectively. Because of predator-prey relationships, water conditions, and size only some types of fishes can survive with some other types of fishes in the same tank.
PROBLEM DEFINITION:
Type A B C D E F
Cannot be with B, C A ,C A, B, D,
E B,C C, F E
The following table gives information about the fishes that cannot be together:
Our task is to arrange the fishes in a minimum number of Tanks.
GRAPH CONSTRUCTION
• To model the situation, we simply need to construct a graph in which each vertex represents one of the types of fish and each edge connects vertices that are not compatible.
• The graph thus constructed turns out to be an interval graph.
GRAPH CONSTRUCTION
Vertex : Fish type Edge: Not Compatible A
B
C
D
E
F
RELATION TO A GRAPH PROBLEM
Set of interval’sD F
B EA
C
1 2 3 4 5 6 7 8 9 10
RELATION TO A GRAPH PROBLEM
This Real world problem is converted to
“interval graph coloring problem”.
An “interval graph” is the graph showing
intersecting intervals on a line. So, we
associate a set of intervals I={I1,…,In} on a
line with the interval graph G=(V,E),where
V={1,…,n} and two vertices, x and y, are
linked by an edge if and only if Ix∩Iy≠.
RELATION TO A GRAPH PROBLEM
E(G) = {{vi, vj} | IX ∩ IY ≠ ∅}
From the graph , it is an interval graph
which is an undirected graph formed
from a set of intervals I(1,2….10).
SPECIAL PROPERTY
Umbrella Free Ordering:
For every interval graph there will be an
Umbrella Free-Ordering it states that,
arranging the vertices in an order such that
if there is an edge between two vertices
then any edge that lies between the two
vertices must be adjacent to the right vertex
in the ordering.
SPECIAL PROPERTY
An umbrella-free representation of a
graph G is a concatenation (in any order)
of all its connected component umbrella-
free representations.
UF be an umbrella-free representation of
G, the vertices of two distinct connected
components are not interleaved in UF.
SPECIAL PROPERTY
So ordering of our graph using umbrella-free property is:A,B,D,C,E,F
UMBRELLA FREE-ORDERING
PROBLEM SOLUTION
In this Umbrella Free-Ordering, we
place the colours in a certain order.
We can now color the vertices. We
start by assigning blue to F, then Red
to E.
PROBLEM SOLUTION
The C vertex, which is the next to
be colored, can be colored in blue
or other Color because Blue is
already available color, we
choose it for vertex C. We
continue in this way until we
colored the whole graph.
COLORING USING THE UMBRELLA FREE-ORDERING
D 2 F 1 B 3 E 2A 2
C 1
PROBLEM SOLUTION
Assigning fishes to tanks, where the
compatibility between fishes is
considered. This problem comes
down to colouring the vertices of an
interval graph under the constraint
that two vertices linked by an edge
cannot be of the same colour.
PROBLEM SOLUTION
Vertex : Fish type Edge: Not Compatible A
B
C
D
E
F
PROBLEM SOLUTION
This coloring requires Three colors,
which means we need three tanks to
assign 6 types of fishes.
Blue Tank Red Tank Black Tank
Fetas & Certas Alphas, Deltas & Epsalas
Betas
PROBLEM SOLUTION
A
B
C
D
E
F
COMMENTS
If this is not the case i.e: if the fish combinations
are different interval graph is not constructed an
arbitrary graph is constructed
For arbitrary graphs no body knows correct
algorithms to colour those arbitrary graphs.
REFERENCES http://
www.polymtl.ca/pub/sites/lagrapheur/docs/en/documents/NotesChap3.pdf
http://www.sciencedirect.com/science/article/pii/S0166218X06000862
http://en.wikipedia.org/wiki/Interval_graph
http://arxiv.org/pdf/1004.4560.pdf
REFERENCESTHANK YOU
ANY QUERIES