Several notions of rank-width for countable graphs

Bruno Courcelle (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 has the compactness property if these two values are equal. Our notion of rank-width that uses quasi-trees (trees where paths may have the order type of rational numbers) 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.

Mots clés: Koenig's Lemma; Rank-width; linear rank-width; compactness

Domaines: Logique en informatique [cs.LO]

Bruno Courcelle. Several notions of rank-width for countable graphs. Journal of Combinatorial Theory, Series B, 2017, 123, pp.186-214. ⟨hal-00957618⟩