Nondeficient Sets in Graphs

Let G = (V, E) be a graph without isolated vertices. A subset U subset of V is called a nondeficient set in G if vertical bar N(S)vertical bar >=vertical bar S vertical bar for all S subset of U. The maximum cardinality of a nondeficient set of G is called the nondeficient number of G and is denoted by nd(G). Any nondeficient set U with vertical bar U vertical bar = nd(G) is called a nd-set of G. In this paper we initiate a study of this parameter and determine the nondeficient number of several families of graphs. We characterize graphs G for which V (G) is a nd-set. Also we determine the value nd(G) in terms of critical independence number of G. Further we obtain lower and upper bounds for nd(G) and characterize graphs which attain the upper bound.