Excluded vertex-minors for graphs of linear rank-width at most k

Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite obstruction set O-k of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in O-k However, no attempts have been made to bound the number of graphs in O-k for k >= 2. We show that for each k, there are at least 2(Omega(3k)) pairwise locally non-equivalent graphs in O-k, and therefore the number of graphs in O-k is at least double exponential. To prove this theorem, it is necessary to characterize when two graphs in O-k are locally equivalent. A graph is a block graph if all of its blocks are complete graphs. We prove that if two block graphs without simplicial vertices of degree at least 2 are locally equivalent, then they are isomorphic. This not only is useful for our theorem but also implies a theorem of Bouchet (1988) stating that if two trees are locally equivalent, then they are isomorphic. (C) 2014 Elsevier Ltd. All rights reserved.