Abstract

We study properties of the sets of minimal forbidden minors for the families of graphs having a vertex cover of size at most $k$. We denote this set by $\Obs(\VCfam{k})$ and call it the set of obstructions. Our main result is to give a tight vertex bound of $\Obs(\VCfam{k})$, and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for \VCfam{k} and this graph is $C_{2k+1}$. We also find two iterative methods to generate graphs in $\Obs(\VCfam{(k+1)})$ from any graph in $\Obs(\VCfam{k})$.