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.

翻译：若存在字母表V上的词w，使得不同字母x和y在w中交替出现当且仅当$xy \in E$，则称图G(V, E)是可词表示的。本文引入p-完全平方与p-完全无平方可词表示图的概念。若词中任意字母子集上的导出子词均不包含满足$|X| \ge p$的平方XX，则称该词是p-完全无平方的；若图允许这样的表示，则称其为p-完全无平方可词表示图。我们定义了p-完全无平方均匀词表示并研究其性质。我们证明：允许此类表示的图均禁止Kp作为导出子图，且其识别问题对任意p均为NP难问题。针对p=1和2的情况，我们给出了完整的刻画。我们证明了每个$K_p$-自由圆图都允许p-完全无平方均匀表示，且任何3-完全无平方均匀可词表示图的表示数至多为三。最后，我们提出了一种针对p=3情形生成新实例的构造性方法。