On lower bound of edge number of minimal edge 1-extension of starlike tree

For a given graph $G$ with $n$ nodes, we say that graph $G^*$ is its 1-edge extension if for each edge $e$ of $G^*$ the subgraph $G^*-e$ contains graph $G$ up to isomorphism. Graph $G^*$ is minimal 1-edge extension of graph $G$ if $G^*$ has $n$ nodes and there is no 1-edge extension with $n$ nodes of graph $G$ having fewer edges than $G$. A tree is called starlike if it has exactly one node of degree greater than two. We give a lower bound of edge number of minimal edge 1-extension of starlike tree and provide family on which this bound is achieved.