Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Note on Threshold Dimension of Permutation Graphs   A graph $G(V,E)$ is a threshold graph if there exist non-negative reals $w_v,
v \in V$ and $t$ such that for every $U \subseteq V$, $\sum_{v \in U} w_v\leq
t$ if and only if $U$ is a stable set. The {\it threshold dimension} of a graph
$G(V,E)$, denoted as $t(G)$, is the smallest integer $k$ such that $E$ can be
covered by $k$ threshold spanning subgraphs of $G$. A permutation graph is a
graph that can be represented as the intersection graph of a family of line
segments that connect two parallel lines in the Euclidean plane. In this paper
we will show that if $G$ is a permutation graph then $t(G) \leq \alpha(G)$
(where $\alpha(G)$ is the cardinality of maximum independent set in $G$) and
this bound is tight. As a corollary we will show that $t(G) \leq \frac{n}{2}$
where $n$ is the number of vertices in the permutation graph $G$. This bound is
also tight.

4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
19
The categories of this paper are: eess
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The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con