  In this paper we study graphs defined by pattern-avoiding words.
Word-representable graphs have been studied extensively following their
introduction in 2000 and are the subject of a book published by Kitaev in 2015.
Recently there has been interest in studying graphs represented by
pattern-avoiding words. In particular, in 2016, Gao, Kitaev, and Zhang
investigated 132-representable graphs, that is, word-representable graphs that
can be represented by a word which avoids the pattern 132. They proved that all
132- representable graphs are circle graphs and provided examples and
properties of 132-representable graphs. They posed several questions, some of
which we answer in this paper.
  One of our main results is that not all circle graphs are 132-representable,
thus proving that 132-representable graphs are a proper subset of circle
graphs, a question that was left open in the paper by Gao et al. We show that
123-representable graphs are also a proper subset of circle graphs, and are
different from 132-representable graphs. We also study graphs represented by
pattern-avoiding 2-uniform words, that is, words in which every letter appears
exactly twice.
