Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On bounds on bend number of split and cocomparability graphs   A path is a simple, piecewise linear curve made up of alternating horizontal
and vertical line segments in the plane. A $k$-bend path is a path made up of
at most $k + 1$ line segments. A $B_k$-VPG representation of a graph is a
collection of $k$-bend paths such that each path in the collection represents a
vertex of the graph and two such paths intersect if and only if the vertices
they represent are adjacent in the graph. The graphs that have a $B_k$-VPG
representation are called $B_k$-VPG graphs. It is known that the poset
dimension $dim(G)$ of a cocomparability graph $G$ is greater than or equal to
its bend number $bend(G)$. Cohen et al. ({\textsc{order 2015}}) asked for
examples of cocomparability graphs with low bend number and high poset
dimension. We answer this question by proving that for each $m, t \in
\mathbb{N}$, there exists a cocomparability graph $G_{t,m}$ with $t <
bend(G_{t,m}) \leq 4t+29$ and $dim(G_{t,m})-bend(G_{t,m})>m$. Techniques used
to prove the above result, allows us to partially address the open question
posed by Chaplick et al. ({\textsc{wg 2012}}) who asked whether
$B_k$-VPG-chordal $\subsetneq$ $B_{k+1}$-VPG-chordal for all $k \in
\mathbb{N}$. We address this by proving that there are infinitely many $m \in
\mathbb{N}$ such that $B_m$-VPG-split $\subsetneq$ $B_{m+1}$-VPG-split which
provides infinitely many positive examples. We use the same techniques to prove
that, for all $t \in \mathbb{N}$, $B_t$-VPG-$Forb(C_{\geq 5})$ $\subsetneq$
$B_{4t+29}$-VPG-$Forb(C_{\geq 5})$, where $Forb(C_{\geq 5})$ denotes the family
of graphs that does not contain induced cycles of length greater than 4.
Furthermore, we show that for all $t \in \mathbb{N}$, $PB_t$-VPG-split
$\subsetneq PB_{36t+80}$-VPG-split, where $PB_t$-VPG denotes the class of
graphs with proper bend number at most $t$.

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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom