On $k\textrm{-}11$-representable graphs

Distinct letters $x$ and $y$ alternate in a word $w$ if after deleting in $w$ all letters but the copies of $x$ and $y$ we either obtain a word of the form $xyxy\cdots$ (of even or odd length) or a word of the form $yxyx\cdots$ (of even or odd length). 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 $E$. Thus, edges of $G$ are defined by avoiding the consecutive pattern 11 in a word representing $G$, that is, by avoiding $xx$ and $yy$. In 2017, Jeff Remmel has introduced the notion of a $k$-11-representable graph for a non-negative integer $k$, which generalizes the notion of a word-representable graph. Under this representation, edges of $G$ are defined by containing at most $k$ occurrences of the consecutive pattern 11 in a word representing $G$. Thus, word-representable graphs are precisely $0$-11-representable graphs. In this paper, we study properties of $k$-11-representable graphs for $k\geq 1$, in particular, showing that the class of word-representable graphs, studied intensively in the literature, is contained strictly in the class of 1-11-representable graphs. Another particular result that we prove is the fact that the class of interval graphs is precisely the class of 1-11-representable graphs that can be represented by uniform words containing two copies of each letter. This result can be compared with the known fact that the class of circle graphs is precisely the class of 0-11-representable graphs that can be represented by uniform words containing two copies of each letter. Also, one of our key results in this paper is the fact that any graph is $k$-11-representable for some $k\geq 0$.