In 1980, Claude Berge introduced B- graphs. These are graphs in which every vertex is contained in a maximum independent set. This idea led to the concept of excellence in graphs which has been introduced by Fricke et al with respect to several parameters like domination, independence, etc. Let be a parameter and let G (V, E) be simple graph. A vertex v in V (G) is said to be -good if v belongs to a -minimum ( -maximum) set of G according as is a super hereditary (hereditary) parameter. v is said to be -bad if it is not -good. A graph G is said to be -excellent if every vertex of G is -good. G is - commandable if number of -good vertices in G is strictly greater than the number -bad vertices of G and there should be at least one -bad vertex in G. G is said to be -fair if number of - good vertices in G is equal to the number of - bad vertices in G and G is said to be -poor if number of -bad vertices in G is strictly greater than the number of - good vertices in G. This book is devoted to the study of graphs with respect to the Independence and vertex covering parameters.

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