p-complete square-free Word-representation of Word-representable Graphs

A graph G(V, E) is word-representable if there exists a word w over V such that distinct letters x and y alternate in w iff $xy \in E$. We introduce p-complete squares and p-complete square-free word-representable graphs. A word is p-complete square-free if no induced subword over any subset of letters contains a square XX with $|X| \ge p$. A graph is p-complete square-free if it admits such a representation. We define p-complete square-free uniform word-representations and study their properties. We show that any graph admitting such a representation forbids Kp as an induced subgraph and that the recognition problem is NP-hard for arbitrary p. For p=1 and 2, we give complete characterisations. We prove that every $K_p$-free circle graph admits a p-complete square-free uniform representation and that any 3-complete square-free uniform word-representable graph has representation number at most three. We present a constructive method for generating new examples for p=3.