Sequential cable constructions and linear rank-width

Abstract

We introduce split-free cable terms and cable plays, a sequential graph-construction language whose live cables impose uniform GF(2)-row behaviour across the current cut. Every play of width w gives a birth-order layout whose cutrank is at most half of w, rounded down, so the sequential split-free width is at least twice the linear rank-width. At the first nontrivial level we prove an exact characterization: a connected graph with at least two vertices has linear rank-width at most one exactly when it admits a stream, equivalently a singleton-birth play of width at most four. We show that unrestricted term width and sequential width differ unboundedly on trees, calibrate the construction on the net graph, and formulate an affine upper-bound conjecture relating sequential split-free width to linear rank-width. For the rank-two case we prove a two-accumulator scheduling criterion that yields width-six plays under a natural future-uniformity hypothesis.