Word-representable graphs from a word’s perspective

Word-representable graphs were introduced in 2008 by Kitaev and Pyatkin in the context of semigroup theory. Graphs are called word-representable if there exists a word with the graph’s nodes as letters such that the letters in the word alternate iff there is an edge between them in the graph. Until today numerous works investigated the word-representability of graphs but mostly from the graph perspective. In this work, we change the perspective to the words, i.e., we take classes of words and investigate the represented graphs. Our first subject of interest are the conjugates of words: we determine exactly which graphs are represented if we rotate the word. Afterwards, we look at k-local words introduced by Day et al. (FSTTCS LIPIcs, 2017) in order to gain more insights into this class of words. Here, we investigate especially which graphs are represented by 1-local words. Lastly, we prove that the language of all words representing a graph is regular. We were also able to characterise k-representable graphs, solving an open problem.