In the mathematical field of graph theory, a word-representable graph is a graph that can be characterized by a word (or sequence) whose entries alternate in a prescribed way. In particular, if the vertex set of the graph is V, one should be able to choose a word w over the alphabet V such that letters a and b alternate in w if and only if the pair ab is an edge in the graph. (Letters a and b alternate in w if, after removing from w all letters but the copies of a and b, one obtains a word abab... or a word baba....) For example, the cycle graph labeled by a, b, c and d in clock-wise direction is word-representable because it can be represented by abdacdbc: the pairs ab, bc, cd and ad alternate, but the pairs ac and bd do not. The word w is G's word-representant, and one says that that w represents G. The smallest (by the number of vertices) non-word-representable graph is the wheel graph W5, which is the only non-word-representable graph on 6 vertices. The definition of a word-representable graph works both in labelled and unlabelled cases since any labelling of a graph is equivalent to any other labelling. Also, the class of word-representable graphs is hereditary. Word-representable graphs generalise several important classes of graphs such as circle graphs, 3-colorable graphs and comparability graphs. Various generalisations of the theory of word-representable graphs accommodate representation of any graph.
History Word-representable graphs were introduced by Sergey Kitaev in 2004 based on joint research with Steven Seif on the Perkins semigroup, which has played an important role in semigroup theory since 1960. The first systematic study of word-representable graphs was undertaken in a 2008 paper by Kitaev and Artem Pyatkin, starting development of the theory. One of key contributors to the area is Magnús M. Halldórsson. Up to date, 35+ papers have been written on the subject, and the core of the book by Sergey Kitaev and Vadim Lozin is devoted to the theory of word-representable graphs. A quick way to get familiar with the area is to read one of the survey articles.
Motivation to study the graphs According to, word-representable graphs are relevant to various fields, thus motivating to study the graphs. These fields are algebra, graph theory, computer science, combinatorics on words, and scheduling. Word-representable graphs are especially important in graph theory, since they generalise several important classes of graphs, e.g. circle graphs, 3-colorable graphs and comparability graphs.
Early results It was shown in that a graph G is word-representable if it is k-representable for some k, that is, G can be represented by a word having k copies of each letter. Moreover, if a graph is k-representable then it is also (k + 1)-representable. Thus, the notion of the representation number of a graph, as the minimum k such that a graph is word-representable, is well-defined. Non-word-representable graphs have the representation number ∞. Graphs with representation number 1 are precisely the set of complete graphs, while graphs with representation number 2 are precisely the class of circle non-complete graphs. In particular, forests (except for single trees on at most 2 vertices), ladder graphs and cycle graphs have representation number 2. No classification for graphs with representation number 3 is known. However, there are examples of such graphs, e.g. Petersen's graph and prisms. Moreover, the 3-subdivision of any graph is 3-representable. In particular, for every graph G there exists a 3-representable graph H that contains G as a minor. A graph G is permutationally representable if it can be represented by a word of the form p1p2...pk, where pi is a permutation. On can also say that G is permutationally k-representable. A graph is permutationally representable iff it is a comparability graph. A graph is word-representable implies that the neighbourhood of each vertex is permutationally representable (i.e. is a comparability graph). Converse to the last statement is not true. However, the fact that the neighbourhood of each vertex is a comparability graph implies that the Maximum Clique problem is polynomially solvable on word-representable graphs.
Semi-transitive orientations Semi-transitive orientations provide a powerful tool to study word-representable graphs. A directed graph is semi-transitively oriented iff it is acyclic and for any directed path u1→u2→ ...→ut, t ≥ 2, either there is no edge from u1 to ut or all edges ui → uj exist for 1 ≤ i < j ≤ t. A key theorem in the theory of word-representable graphs states that a graph is word-representable iff it admits a semi-transitive orientation. As a corollary to the proof of the key theorem one obtain an upper bound on word-representants: Each non-complete word-representable graph G is 2(n − κ(G))-representable, where κ(G) is the size of a maximal clique in G. As an immediate corollary of the last statement, one has that the recognition problem of word-representability is in NP. In 2014, Vincent Limouzy observed that it is an NP-complete problem to recognise whether a given graph is word-representable. Another important corollary to the key theorem is that any 3-colorable graph is word-representable. The last fact implies that many classical graph problems are NP-hard on word-representable graphs.
Overview of selected results
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