Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 2-uniform words: cycle graphs, and an algorithm to verify specific
  word-representations of graphs   For an arbitrary word $w$ on an alphabet, we can define the alternating
symbol graph, $G(w)$, as the graph in which the edge $(a, b)$ is in $E$ iff the
letters $a$ and $b$ alternate in the word $w$. A graph $G = (V, E)$ is said to
be word-representable if $G = G(w)$ for some word $w$ on $V$.
  The general problem of checking whether a graph is word-representable has
been shown to be NP-complete. However, checking whether a given graph is a
2-uniform word-representable (each letter occurring exactly twice in the word)
has an $O(V^2)$-time algorithm, described by Spinrad. Related to this, we
propose a novel $O(V \log(V) + E)$ time algorithm implementing Fenwick Trees to
check whether $G(w) = G$, for a given 2-uniform word $w$ and a graph $G = (V,
E)$.
  We also prove that the number of 2-uniform words representing the labelled
$n$-vertex cycle graphs is precisely $4n$.

29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc