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.

翻译：设图$G = (\{1, 2, \ldots, n\}, E)$是$12$-可表示的，如果存在一个由$\{1, 2, \ldots, n\}$构成的词$w$，使得对于满足$i < j$的两个顶点$i$和$j$，它们在图中相邻当且仅当每个$j$在$w$中都出现在每个$i$之前。这类图已被证明等价于简单三角图的补图。这一等价性提供了基于顶点排序中禁止模式的刻画方法以及多项式时间识别算法。$12$-可表示图类由Jones等人(2015)作为词可表示图的一种变体引入。Kitaev和Lozin(2015)建议的词可表示图的一个通用研究方向是研究由特定类型词表示的图。例如，Gao、Kitaev和Zhang(2017)以及Mandelshtam(2019)研究了由避免模式的词表示的词可表示图。遵循这一研究方向，本文研究由避免模式$p$的词表示的$12$-可表示图。当$p$长度为$2$时，此类图是平凡的。当$p = 111$、$121$、$231$和$321$时，此类图类等价于众所周知图类，例如平凡完美图与二分置换图。对于$p = 123$、$132$和$211$的情况，本文提供了禁止模式的刻画。