Abstract
We present a characterization of the linear rank-width of distance-hereditary graphs. Using the characterization, we show that the linear rank-width of every \(n\)-vertex distance-hereditary graph can be computed in time \(\mathcal {O}(n^2\cdot \log (n))\), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical split decompositions with an idea of [Megiddo, Hakimi, Garey, Johnson, Papadimitriou: The complexity of searching a graph. JACM 1988], used for computing the path-width of trees. We also provide a set of distance-hereditary graphs which contains the set of distance-hereditary vertex-minor obstructions for linear rank-width. The set given in [Jeong, Kwon, Oum: Excluded vertex-minors for graphs of linear rank-width at most k. STACS 2013: 221–232] is a subset of our obstruction set.
Isolde Adler: Supported by the German Research Council, Project GalA, AD 411/1-1.
Mamadou Moustapha Kanté: Supported by the French Agency for Research under the DORSO project.
O-joung Kwon: Supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2011-0011653).
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Adler, I., Kanté, M.M., Kwon, Oj. (2014). Linear Rank-Width of Distance-Hereditary Graphs. In: Kratsch, D., Todinca, I. (eds) Graph-Theoretic Concepts in Computer Science. WG 2014. Lecture Notes in Computer Science, vol 8747. Springer, Cham. https://doi.org/10.1007/978-3-319-12340-0_4
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