In the present paper the author introduces the notion of \(\varepsilon\)- girth of plane compacta \(E\) with respect to a straight line \(\ell_ \theta\), which forms an angle \(\theta\) with the real axis. The author proves the next theorems. Theorem 1. \(E\) are \(N\)-compacta (i.e. compacta, the complement of which to \(\overline\mathbb{C}\) is a minimal domain in the sense of Koebe at the map on the plane with straight line cuts being parallel to the real axis) iff \(E\) satisfies a condition of \(\varepsilon\)-girth with respect to \(\ell_ \theta\). Theorem 3. \(E\) is an NED-set (i.e. removable set for the class of regular functions with bounded Dirichlet integral) if \(E\) satisfies a condition of \(\varepsilon\)-girth with respect to \(\ell_{\theta 4}\) \(\ell_{\pi/2}\).