Defining Sets of Projective Planes and Biplanes and Their Residuals

Suppose \(S\) is a defining set of a symmetric \(2\)-( \(v, k, \lambda\) ) design \(D\), where \(\lambda = 1\) or \(2\); that is, \(D\) is a projective plane or a biplane.
In this paper, conditions under which the residual of \(S\) is a defining set of the residual of \(D\) are investigated.
As a consequence, inequalities relating the sizes of smallest defining sets of \(D\) and of the residual of \(D\) are obtained.
The exact sizes of smallest defining sets of \({PG}(2, 5)\), \({AG}(2, 5)\), and the three non-isomorphic \(2\)-( \(10, 4, 2\) ) designs are determined.