Several notions of rank-width for countable graphs

We define several notions of rank-width for countable graphs. We compare, for each of them the width of a countable graph with the least upper bound of the widths of its finite induced subgraphs. A width measure has the compactness property if these two values are equal. One of our notions of rank-width for countable graphs uses quasi-trees (generalized trees where “paths” may have any order type) and has this property. So has linear rank-width, based on arbitrary linear orders. A more natural notion of rank-width based on countable cubic trees (we call it discrete rank-width) has a weaker type of compactness: the corresponding width is at most twice the least upper bound of the widths of the finite induced subgraphs. The notion of discrete linear rank-width, based on discrete linear orders has no compactness property.