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 $\mathcal{O}_k$ 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 $\mathcal{O}_k$. However, no attempts have been made to bound the number of graphs in $\mathcal{O}_k$ for $k\ge 2$. We show that for each $k$, there are at least $2^{Ω(3^k)}$ pairwise locally non-equivalent graphs in $\mathcal{O}_k$, and therefore the number of graphs in $\mathcal{O}_k$ is at least double exponential. To prove this theorem, it is necessary to characterize when two graphs in $\mathcal O_k$ 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 [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.

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