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

Abstract

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 $4n$.