  Given a finite word $w$ over a finite alphabet $V$, consider the graph with
vertex set $V$ and with an edge between two elements of $V$ if and only if the
two elements alternate in the word $w$. Such a graph is said to be
word-representable or 11-representable by the word $w$; this latter terminology
arises from the phenomenon that the condition of two elements $x$ and $y$
alternating in a word $w$ is the same as the condition of the subword of $w$
induced by $x$ and $y$ avoiding the pattern 11. In this paper, we first study
minimal length words which word-represent graphs, giving an explicit formula
for both the length and the number of such words in the case of trees and
cycles. We then extend the notion of word-representability (or
11-representability) of graphs to $t$-representability of graphs, for any
pattern $t$ on two letters. We prove that every graph is $t$-representable for
any pattern $t$ on two letters (except for possibly one class of $t$). Finally,
we pose a few open problems for future consideration.
