  The notion of a $k$-11-representable graph was introduced by Jeff Remmel in
2017 and studied by Cheon et al.\ in 2019 as a natural extension of the
extensively studied notion of word-representable graphs, which are precisely
0-11-representable graphs. A graph $G$ is $k$-11-representable if it can be
represented by a word $w$ such that for any edge (resp., non-edge) $xy$ in $G$
the subsequence of $w$ formed by $x$ and $y$ contains at most $k$ (resp., at
least $k+1$) pairs of consecutive equal letters. A remarkable result of Cheon
at al.\ is that {\em any} graph is 2-11-representable, while it is unknown
whether every graph is 1-11-representable. Cheon et al.\ showed that the class
of 1-11-representable graphs is strictly larger than that of word-representable
graphs, and they introduced a useful toolbox to study 1-11-representable
graphs.
  In this paper, we introduce new tools for studying 1-11-representation of
graphs. We apply them for establishing 1-11-representation of Chv\'{a}tal
graph, Mycielski graph, split graphs, and graphs whose vertices can be
partitioned into a comparability graph and an independent set.
