On minimum $ (K_{1,r};k) $-vertex stable graphs on the exact number of vertices

A graph $G$ is said to be $(H;k)$-vertex stable if $G$ contains a~subgraph isomorphic to $H$ even after removing any $k$ of its vertices alongside with their incident edges. We will denote by $\text{stab}(H;k)$ the minimum size among sizes of all $(H;k)$-vertex stable graphs. In this paper we consider a~case where the structure $H$ is a~star graph $K_{1,r}$ and the the number of vertices in $G$ is exact, \ie equal to $1 + r + k$. We will show that under the above assumptions $\text{stab}(K_{1,r};k)$ equals either $\frac{1}{2}(k + 1)(2r + k)$, $\frac{1}{2}\big((r + k)^{2} - 1\big)$ or $\frac{1}{2}(r + k)^{2}$. Moreover, we will characterize all the extremal graphs.