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For an arbitrary word $w$ on an alphabet, we can define the alternatingsymbol graph, $G(w)$, as the graph in which the edge $(a, b)$ is in $E$ iff theletters $a$ and $b$ alternate in the word $w$. A graph $G = (V, E)$ is said tobe word-representable if $G = G(w)$ for some word $w$ on $V$.  The general problem of checking whether a graph is word-representable hasbeen shown to be NP-complete. However, checking whether a given graph is a2-uniform word-representable (each letter occurring exactly twice in the word)has an $O(V^2)$-time algorithm, described by Spinrad. Related to this, wepropose a novel $O(V \log(V) + E)$ time algorithm implementing Fenwick Trees tocheck 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 $4n$.

2-uniform words: cycle graphs, and an algorithm to verify specific word-representations of graphs