Properties of vertex cover obstructions

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 OO(k-VERTEX COVER) and call it the set of obstructions. Our main result is to give a tight vertex bound of OO(k-VERTEX COVER), and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for k-VERTEX COVER and this graph is C2k+1C2k+1. We also find two iterative methods to generate graphs in OO((k+1)(k+1)-VERTEX COVER) from any graph in OO(k-VERTEX COVER).