Semi-transitive orientations and word-representable graphs

Magnús M. Halldórsson, Sergey Kitaev, Artem Pyatkin

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)
103 Downloads (Pure)


A graph G=(V,E) is a \emph{word-representable graph} if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x,y)∈E for each x≠y. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of G is the minimum k such that G is a representable by a word, where each letter occurs k times; such a k exists for any word-representable graph. We show that the representation number of a word-representable graph on n vertices is at most 2n, while there exist graphs for which it is n/2.
Original languageEnglish
Pages (from-to)164-171
Number of pages14
JournalDiscrete Applied Mathematics
Early online date24 Aug 2015
Publication statusPublished - 11 Mar 2016


  • graphs
  • comparability graphs
  • circle graphs
  • complexity
  • word-representability
  • orientations


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