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A graph <jats:italic>G<\/jats:italic> is <jats:italic>k<\/jats:italic>-11-representable if it can be represented by a word <jats:italic>w<\/jats:italic> such that for any edge (resp., non-edge) <jats:italic>xy<\/jats:italic> in <jats:italic>G<\/jats:italic> the subsequence of <jats:italic>w<\/jats:italic> formed by <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic> contains at most <jats:italic>k<\/jats:italic> (resp., at least <jats:inline-formula><jats:alternatives><jats:tex-math>$$k+1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) pairs of consecutive equal letters. A remarkable result of Cheon at al. is that <jats:italic>any<\/jats:italic> 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\u00e1tal graph, Mycielski graph, split graphs, and graphs whose vertices can be partitioned into a comparability graph and an independent set.<\/jats:p>","DOI":"10.1007\/s00373-024-02825-1","type":"journal-article","created":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T01:01:46Z","timestamp":1723597306000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["New Tools to Study 1-11-Representation of Graphs"],"prefix":"10.1007","volume":"40","author":[{"given":"Mikhail","family":"Futorny","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7100-9596","authenticated-orcid":false,"given":"Sergey","family":"Kitaev","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Artem","family":"Pyatkin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,14]]},"reference":[{"key":"2825_CR1","first-page":"724","volume":"274","author":"A Bouchet","year":"1972","unstructured":"Bouchet, A.: Caract\u00e9risation des symboles crois\u00e9s de genre nul. 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