  A graph $G = (\{1, 2, \ldots, n\}, E)$ is $12$-representable if there is a
word $w$ over $\{1, 2, \ldots, n\}$ such that two vertices $i$ and $j$ with $i
< j$ are adjacent if and only if every $j$ occurs before every $i$ in $w$.
These graphs have been shown to be equivalent to the complements of
simple-triangle graphs. This equivalence provides a characterization in terms
of forbidden patterns in vertex orderings as well as a polynomial-time
recognition algorithm. The class of $12$-representable graphs was introduced by
Jones et al. (2015) as a variant of word-representable graphs. A general
research direction for word-representable graphs suggested by Kitaev and Lozin
(2015) is to study graphs representable by some specific types of words. For
instance, Gao, Kitaev, and Zhang (2017) and Mandelshtam (2019) investigated
word-representable graphs represented by pattern-avoiding words. Following this
research direction, this paper studies $12$-representable graphs represented by
words that avoid a pattern $p$. Such graphs are trivial when $p$ is of length
$2$. When $p = 111$, $121$, $231$, and $321$, the classes of such graphs are
equivalent to well-known classes, such as trivially perfect graphs and
bipartite permutation graphs. For the cases where $p = 123$, $132$, and $211$,
this paper provides forbidden pattern characterizations.
