  The notion of a word-representable graph has been studied in a series of
papers in the literature. A graph $G=(V,E)$ is word-representable if there
exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate
in $w$ if and only if $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is
equivalent to saying that $G$ is word-representable if for all $x,y \in \{1,
\ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$
consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive
occurrence of the pattern 11.
  In this paper, we introduce the study of $u$-representable graphs for any
word $u \in \{1,2\}^*$. A graph $G$ is $u$-representable if and only if there
is a labeled version of $G$, $G=(\{1, \ldots, n\}, E)$, and a word $w \in \{1,
\ldots, n\}^*$ such that for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and
only if $w_{\{x,y\}}$ has no consecutive occurrence of the pattern $u$. Thus,
word-representable graphs are just $11$-representable graphs. We show that for
any $k \geq 3$, every finite graph $G$ is $1^k$-representable. This contrasts
with the fact that not all graphs are 11-representable graphs.
  The main focus of the paper is the study of $12$-representable graphs. In
particular, we classify the $12$-representable trees. We show that any
$12$-representable graph is a comparability graph and the class of
$12$-representable graphs include the classes of co-interval graphs and
permutation graphs. We also state a number of facts on $12$-representation of
induced subgraphs of a grid graph.
