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Semi-Transitive Orientations and Word-Representable Graphs
A graph is a \emph{word-representable graph} if there exists a word
over the alphabet such that letters and alternate in if and
only if for each .
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 is the minimum such that is a
representable by a word, where each letter occurs times; such a exists
for any word-representable graph. We show that the representation number of a
word-representable graph on vertices is at most , while there exist
graphs for which it is .Comment: arXiv admin note: text overlap with arXiv:0810.031
On word-representability of polyomino triangulations
A graph is word-representable if there exists a word over the
alphabet such that letters and alternate in if and only if
is an edge in . Some graphs are word-representable, others are not.
It is known that a graph is word-representable if and only if it accepts a
so-called semi-transitive orientation.
The main result of this paper is showing that a triangulation of any convex
polyomino is word-representable if and only if it is 3-colorable. We
demonstrate that this statement is not true for an arbitrary polyomino. We also
show that the graph obtained by replacing each -cycle in a polyomino by the
complete graph is word-representable. We employ semi-transitive
orientations to obtain our results
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