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 O ( k-V ERTEX C OVER) and call it the set of obstructions. Our main result is to give a tight vertex bound of O ( k-V ERTEX C OVER), and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for k-V ERTEX C OVER and this graph is C 2 k + 1 . We also find two iterative methods to generate graphs in O ( ( k + 1 ) -V ERTEX C OVER) from any graph in O ( k-V ERTEX C OVER).
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