International Journal of Pure and Applied Mathematics Volume 87 No. 6 2013, 763-769 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v87i6.5 P A ijpam.eu SUPER STRONGLY PERFECTNESS OF SOME GRAPHS R. Mary Jeya Jothi 1 , A. Amutha 2 1,2 Department of Mathematics Sathyabama University Chennai, 119, INDIA Abstract: A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H . The structure of Super Strongly Perfect Graphs have been charac- terized by some classes of graphs like Cycle graphs, Circulant graphs, Complete graphs, Complete Bipartite graphs etc., In this paper, we have analysed some other graph classes like, Bicyclic graphs, Dumb bell graphs and Star graphs to characterize the structure of Super Strongly Perfect Graphs in a different way. By this we found the cardinality of minimal dominating set and maximal cliques of the above graphs. AMS Subject Classification: 05C75 Key Words: super strongly perfect graph, minimal dominating set, bicyclic graph, dumb bell graph 1. Introduction In this paper, graphs are finite and simple, that is, they have no loops or multiple edges. Let G =(V,E) be a graph. A clique in G is a set X ⊆ V (G) of pair wise adjacent vertices. A subset D of V (G) is called a dominating set if every vertex in V − D is adjacent to at least one vertex in D. A subset S of V is said to be a minimal dominating set if S −{u} is not a dominating set for Received: September 6, 2013 c  2013 Academic Publications, Ltd. url: www.acadpubl.eu