Containment problem for the quasi star configurations of points in $\mathbb{P}^2$

In this paper, the containment problem for the defining ideal of a special type of zero dimensional subschemes of $\mathbb{P}^2$, so called quasi star configurations, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number $0 < \varepsilon < \frac{1}{2}$, we construct an infinite family of homogeneous radical ideals of points in $\mathbb{K}[\mathbb{P}^2]$ such that their resurgences lie in the interval $[2- \varepsilon ,2)$. Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that, %the defining ideal of a quasi star configuration, and all of them have linear resolution.