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In my Ph. D thesis, we derive a structure theorem that helps to determine whether a  plane triangulation  (or a plane near-triangulation) is perfect. This leads to a quadratic time algorithm for checking perfectness of plane triangulations (and plane near-triangulations). For the classes  of plane triangulations and near-triangulations, the present algorithm is an improvement on the cubic complexity algorithm known earlier, discovered by Hsu (1987) that can test whether an arbitrary planar graph is perfect. The structure theorem states that a plane triangulation  (or plane near-triangulation) fails to be perfect if and only if there exists a vertex, an edge or a triangle in the graph,  the neighborhood of which has an odd hole as its boundary.  Thesis Link.

 In my M.Tech thesis, the linear-programming dual of the dominating set problem, called the vertex  packing  problem, is studied. We present a simple proof that shows that the problem does not admit an O(|V|^{1/2−ε}) factor approximation algorithm, for any ε >0, unless NP=ZPP. (Thesis Link)

Email: shemi.nazir @gmail.com

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