Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the bend number of circular-arc graphs as edge intersection graphs of
  paths on a grid   Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can
be represented as the edge intersection graph of paths on a grid (EPG graph),
i.e., one can associate with each vertex of the graph a nontrivial path on a
rectangular grid such that two vertices are adjacent if and only if the
corresponding paths share at least one edge of the grid. For a nonnegative
integer $k$, $B_k$-EPG graphs are defined as EPG graphs admitting a model in
which each path has at most $k$ bends. Circular-arc graphs are intersection
graphs of open arcs of a circle. It is easy to see that every circular-arc
graph is a $B_4$-EPG graph, by embedding the circle into a rectangle of the
grid. In this paper, we prove that every circular-arc graph is $B_3$-EPG, and
that there exist circular-arc graphs which are not $B_2$-EPG. If we restrict
ourselves to rectangular representations (i.e., the union of the paths used in
the model is contained in a rectangle of the grid), we obtain EPR (edge
intersection of path in a rectangle) representations. We may define $B_k$-EPR
graphs, $k\geq 0$, the same way as $B_k$-EPG graphs. Circular-arc graphs are
clearly $B_4$-EPR graphs and we will show that there exist circular-arc graphs
that are not $B_3$-EPR graphs. We also show that normal circular-arc graphs are
$B_2$-EPR graphs and that there exist normal circular-arc graphs that are not
$B_1$-EPR graphs. Finally, we characterize $B_1$-EPR graphs by a family of
minimal forbidden induced subgraphs, and show that they form a subclass of
normal Helly circular-arc graphs.

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The categories of this paper are: acc-phys
33
The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: supr-con
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg