Universal 1-11-representability of graphs

Determine whether every finite graph is 1-11-representable.

Background

A graph is 1-11-representable if it admits a word representation in which the subsequence induced by each edge contains at most one occurrence of two consecutive equal letters, while the subsequence induced by each non-edge contains at least two such occurrences. The paper establishes this property for all graphs with at most eight vertices, extending the previously known bound of seven vertices, but does not resolve the question for arbitrary graphs.