The literature on word-representable graphs is quite rich, and a number of variations of the original definition have been proposed over the years. We are initiating a systematic study of such variations based on formal languages. In our framework, we can associate a graph class to each language over the binary alphabet \{0,1\}. All graph classes that are language-representable in this sense are hereditary and enjoy further common properties. Besides word-representable graphs and, more generally, 1^k- or k-11-representable graphs, we can identify many more graph classes in our framework, like (co)bipartite graphs, (co)comparability graphs, to name a few. It was already known that any graph is 111- or 2-11-representable. When such representations are considered for storing graphs, 111- or 2-11-representability bears the disadvantage of being significantly inferior to standard adjacency matrices or lists. We prove that quite famous languages like the palindromes, the copy language or the Lyndon words can match the efficiency of standard graph representations. The perspective of language theory allows us to prove general results that hold for all graph classes that can be defined in this way. This includes certain closure properties (e.g., all language-definable graph classes are hereditary) as well as certain limitations (e.g., all language-representable graph classes contain graphs of arbitrarily large treewidth and of arbitrarily large degeneracy, except a trivial case). As each language describes a graph class, we can also ask decidability questions concerning graph classes, given a concrete presentation of a formal language. We also present a systematic study of graph classes that can be represented by languages in which each letter occurs at most twice. Here, we find graph classes like interval, permutation, circle, bipartite chain, convex, and threshold graphs.

翻译：关于词可表示图的文献相当丰富，多年来已对原始定义提出了多种变体。我们基于形式语言对此类变体展开系统研究。在我们的框架中，可以为二进制字母表{0,1}上的每种语言关联一个图类。所有在此意义上语言可表示的图类都具有遗传性，并享有进一步的共同性质。除词可表示图及更一般的1^k-或k-11可表示图外，我们还能在框架中识别更多图类，例如（余）二分图、（余）可比图等。已知任意图都是111-或2-11可表示的。当此类表示用于存储图时，111-或2-11可表示性存在明显劣于标准邻接矩阵或邻接表的缺陷。我们证明回文语言、复制语言或Lyndon词等著名语言能达到标准图表示的效率。语言理论的视角使我们能够证明适用于所有以此方式定义的图类的一般性结果，包括某些封闭性质（例如所有语言可定义的图类都具有遗传性）以及某些局限性（例如除平凡情况外，所有语言可表示的图类都包含任意大树宽和任意大退化度的图）。由于每种语言描述一个图类，给定形式语言的具体表示，我们还可以探讨关于图类的可判定性问题。我们还系统研究了可由每个字母至多出现两次的语言表示的图类，在此发现了区间图、置换图、圆图、二分链图、凸图和阈值图等图类。