Tree pivot-minors and linear rank-width
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Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree T, the class of graphs that do not contain T as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role from tree-width and path-width. As such, it is natural to examine if for every tree T, the class of graphs that do not contain T as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever T is a tree that is not a caterpillar. We conjecture that the statement is true if T is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree T, the class of T-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if T is a caterpillar; (2) for every caterpillar T on at most four vertices, the class of T-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider T=P_4 and T=K_1,3, but we follow a general strategy: first we show that the class of T-pivot-minor-free graphs is contained in some class of (H_1,H_2)-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of (K_3,S_1,2,2)-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
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