Linear rank-width is a linearized variation of rank-width, and it is deeply
related to matroid path-width. In this paper, we show that the linear
rank-width of every n-vertex distance-hereditary graph, equivalently a graph
of rank-width at most 1, can be computed in time O(n2⋅log2n), and a linear layout witnessing the linear rank-width can be computed with
the same time complexity. As a corollary, we show that the path-width of every
n-element matroid of branch-width at most 2 can be computed in time
O(n2⋅log2n), provided that the matroid is given by an
independent set oracle.
To establish this result, we present a characterization of the linear
rank-width of distance-hereditary graphs in terms of their canonical split
decompositions. This characterization is similar to the known characterization
of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex
separation and search number of a graph. Inf. Comput., 113(1):50--79, 1994].
However, different from forests, it is non-trivial to relate substructures of
the canonical split decomposition of a graph with some substructures of the
given graph. We introduce a notion of `limbs' of canonical split
decompositions, which correspond to certain vertex-minors of the original
graph, for the right characterization.Comment: 28 pages, 3 figures, 2 table. A preliminary version appeared in the
proceedings of WG'1