Distributing Chemical Compounds in Refrigerators using interval graphs By Sairam sista

Contents Problem statement Graph construction Relation to Graph theory Graph definition Special property Graph solution References

Problem Statement Suppose that C1, C2, · · · , Cn are chemical compounds which must be refrigerated under closely monitored conditions. If a compound Ci (i=1 to n) must be kept at a constant temperature between ti and ti’ degrees then how many refrigerators will be needed to store all the compounds? Our task here is to arrange the chemical compounds in minimum number of refrigerators.

Problem Statement Chemical compound Temperatures A 0-25 B 5-15 C 10-50 20-45 E 30-40 F 35-55 G 55-60

Graph construction If two different chemical compounds occurred together in the same refrigerator, then their temperatures must have overlapped. To model the situation, we simply need to construct a graph in which : each vertex represents one of the chemical compounds each edge connects vertices where their respective temperatures overlap.

Graph construction Let us consider a graph G and the vertices from A-G which represent chemical compounds and their edges which represent the overlap of their temperature intervals.

Graph construction

Relation to Graph theory This Real world problem is converted to “interval graph 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 their temperature intervals overlap.The following is the interval graph formed from the graph .

Relation to graph theory

Graph Definition Interval graphs: In a Graph theory,an interval graph is the intersection graph of a family of intervals on the real line. It has one vertex for each interval in the family, and an edge between every pair of vertices whenever intervals intersect. EXAMPLE:

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 Minimum clique cover: To calculate minimum number of cliques in an umbrella free ordering. It is chordal and its complement G is a comparability graph. AT ( asteroidal triple) free. It contains no induced C4 and G is transitively orientable.

Special Property

Graph solution According to the graph with vertices A-G we draw umbrella free ordering from left to right. We then find cliques among the vertices. Umbrella free ordering after assigning clique:

Graph solution

Graph solution From the umbrella free ordering we get 3 cliques. A,B,C and D,E,F and G are the 3 cliques . So here minimum clique cover is 3 Seven chemical compounds (A-G) can here by arranged in 3 refrigerators.

Graph solution Chemical compounds Temperatures Refrigerator temperature A 0-25 15 B 5-15 C 10-50 D 20-45 35 E 30-40 F 35-55 G 55-60 57

NP-complete In computational complexity theory, finding a minimum clique cover is a graph- theoretical NP-complete problem. The problem was one of Richard Karp's original 21 problems shown NP-complete in his 1972 paper "Reducibility Among Combinatorial Problems".

References http://www.csee.umbc.edu/~stephens/NUM/PR OJECTS/refrigerator.html https://halshs.archives-ouvertes.fr/halshs- 00123607/document http://www.researchgate.net/publication/228574 261_On-line_algorithm_for_the_minimal_b- clique_cover_problem_in_interval_graphs http://infoscience.epfl.ch/record/118668/files/EP FL_TH4090.pdf http://worldwidescience.org/topicpages/m/mini mum-diameter+clique+trees.html

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