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 Ok 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 Ok. However, no attempts have been made to bound the number of graphs in Ok for k ≥ 2. We show that for each k, there are at least 2Ω(3 ) pairwise locally non-equivalent graphs in Ok, and therefore the number of graphs in Ok is at least double exponential. To prove this theorem, it is necessary to characterize when two graphs in Ok 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 [Transforming trees by successive local complementations, J. Graph Theory 12 (1988), no. 2, 195–207] stating that if two trees are locally equivalent, then they are isomorphic.