ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2018, Vol. 12, No. 2, pp. 278–296.
Pleiades Publishing, Ltd., 2018.
Original Russian Text
S.V. Kitaev, A.V. Pyatkin, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 2, pp. 19–53.
Word-Representable Graphs: a Survey
S. V. Kitaev
and A. V. Pyatkin
University of Strathclyde, Livingstone Tower, 26 Richmond St., Glasgow, G1 1XH, UK
Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia
Received August 11, 2017; in ﬁnal form, December 12, 2017
Abstract—Letters x and y alternate in a word w if after deleting all letters but x and y in w we
get either a word xyxy . . . or a word yxyx . . . (each of these words can be of odd or even length).
AgraphG =(V,E) is word-representable if there is a ﬁnite word w over an alphabet V such that
the letters x and y alternate in w if and only if xy ∈ E. The word-representable graphs include many
important graph classes, in particular, circle graphs, 3-colorable graphs and comparability graphs.
In this paper we present the full survey of the available results on the theory of word-representable
graphs and the most recent achievements in this ﬁeld.
Keywords: representation of graphs, orientation, word, pattern
The theory of word-representable graphs is a young but promising research area. This theory was
ﬁrst introduced by Kitaev, motivated by the article  studying the Perkins semigroups that play an
important role in the semigroup theory since 1960, in particular, as a tool for constructing examples and
counterexamples . But the ﬁrst systematic study of word-representable graphs was made in , where
this theory had started. Nowadays, there are about 20 papers in this area and also a book by Kitaev and
Lozin  which is devoted to the theory of word-representable graphs. Also, there is a software made by
Glen  that makes a work with word-representable graphs much easier.
Many examples of applications of word-representable graphs, motivating their research, can be found
in . They have applications in algebra, graph theory, computer science, combinatorics, and scheduling
theory. In particular, from the point of view of graph theory they are important because they generalize
several fundamental graph classes (e.g., circle graphs , 3-colorable graphs , and comparability
graphs ). The link with scheduling theory is given in .
The following questions look rather natural:
• Are all graphs word-representable?
• If not then is it possible to characterize the graphs that are or are not word-representable?
• How many graphs are word-representable?
• What is the representation number of a graph, i.e. the minimum length of the word representing
the given graph?
• How (algorithmically) hard is to determine whether the graph is word-representable or not?
• Which graph operations preserve the property of (non-)word-presentability?