Outerplanar partial cubes

Els partial cube-menors son una analogia de la noció de menors als partial cubes. En aquest article determinem el conjunt de pc-menors de les classes dels partial cubes outerplanars i els partial cubes sèrie-paral·lel. Aquest és el primer resultat d’aquest tipus per als partial cubes d’una classe tancada per menors.Partial cube-minors are an analogue of graph minors in partial cubes. We determine the set of forbidden partial cube minors of the classes of outerplanar and series-parallel partial cubes. This is the first result of this type for the partial cubes in a minor closed graph class

Two-dimensional partial cubes

25 pages, 5 figuresWe investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986). Furthermore we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements