204 research outputs found

    Hypercellular graphs: partial cubes without Q3Q_3^- as partial cube minor

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    We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex Q3Q^-_3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

    Polynomial treewidth forces a large grid-like-minor

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    Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ×\ell\times\ell grid minor is exponential in \ell. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} \ell in a graph GG is a set of paths in GG whose intersection graph is bipartite and contains a KK_{\ell}-minor. For example, the rows and columns of the ×\ell\times\ell grid are a grid-like-minor of order +1\ell+1. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least c4logc\ell^4\sqrt{\log\ell} has a grid-like-minor of order \ell. As an application of this result, we prove that the cartesian product GK2G\square K_2 contains a KK_{\ell}-minor whenever GG has treewidth at least c4logc\ell^4\sqrt{\log\ell}.Comment: v2: The bound in the main result has been improved by using the Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte
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