Completing Partial k -Star Designs

A $k$-star is a complete bipartite graph $K_{1,k}$. A partial $k$-star design of order $n$ is a pair $\left(V,A\right)$ where $V$ is a set of $n$ vertices and $A$ is a set of edge-disjoint $k$-stars whose vertex sets are subsets of $V$. If each edge of the complete graph with vertex set $V$ is in some star in $A$, then $\left(V,A\right)$ is a (complete) $k$-star design. We say that $\left(V,A\right)$ is completable if there is a $k$-star design $\left(V,ℬ\right)$ such that $A\subseteq ℬ$. In this paper we determine, for all $k$ and $n$, the minimum number of stars in an uncompletable partial $k$-star design of order $n$.