Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A framework for minimal hereditary classes of graphs of unbounded
  clique-width   We create a framework for hereditary graph classes $\mathcal{G}^\delta$ built
on a two-dimensional grid of vertices and edge sets defined by a triple
$\delta=\{\alpha,\beta,\gamma\}$ of objects that define edges between
consecutive columns, edges between non-consecutive columns (called bonds), and
edges within columns. This framework captures all previously proven minimal
hereditary classes of graph of unbounded clique-width, and many new ones,
although we do not claim this includes all such classes.
  We show that a graph class $\mathcal{G}^\delta$ has unbounded clique-width if
and only if a certain parameter $\mathcal{N}^\delta$ is unbounded. We further
show that $\mathcal{G}^\delta$ is minimal of unbounded clique-width (and,
indeed, minimal of unbounded linear clique-width) if another parameter
$\mathcal{M}^\beta$ is bounded, and also $\delta$ has defined recurrence
characteristics. Both the parameters $\mathcal{N}^\delta$ and
$\mathcal{M}^\beta$ are properties of a triple $\delta=(\alpha,\beta,\gamma)$,
and measure the number of distinct neighbourhoods in certain auxiliary graphs.
  Throughout our work, we introduce new methods to the study of clique-width,
including the use of Ramsey theory in arguments related to unboundedness, and
explicit (linear) clique-width expressions for subclasses of minimal classes of
unbounded clique-width.

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The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-lat
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The categories of this paper are: q-bio
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The categories of this paper are: alg-geom