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Words for the Graphs with Permutation-Representation Number at most Three
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The graphs with permutation-representation number (\textit{prn}) at most two are known. While a characterization for the class of graphs with the \textit{prn} at most three is an open problem, we summarize the graphs of this class that are known so far. Although it is known that the \textit{prn} of trees is at most three, in this work, we devise a polynomial-time algorithm for obtaining a word representing a given tree permutationally. Consequently, we determine the words representing even cycles. Contributing to the class of graphs with the \textit{prn} at most three, we determine the \textit{prn} as well as the representation number of book graphs.
Forward citations
Cited by 2 Pith papers
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Representation Number of Word-Representable Split Graphs
Word-representable split graphs have representation number at most 3, and the graphs with representation number exactly 3 are characterized by a family of induced subgraphs.
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Characterization of Word-Representable Graphs using Modular Decomposition
A graph made by substituting a module into a word-representable graph is word-representable if and only if the module is a comparability graph, yielding a characterization of lexicographic products.
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