Identification, location-domination and metric dimension on interval and permutation graphs. II: Algorithms and complexity
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Publication:2408094
DOI10.1007/S00453-016-0184-1zbMATH Open1371.05212DBLPjournals/algorithmica/FoucaudMNPV17arXiv1405.2424OpenAlexW3101038336WikidataQ56551560 ScholiaQ56551560MaRDI QIDQ2408094
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Publication date: 9 October 2017
Published in: (Search for Journal in Brave)
Abstract: We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted IDENTIFYING CODE, (OPEN) LOCATING-DOMINATING SET and METRIC DIMENSION) of an interval or a permutation graph. In these problems, one asks to distinguish all vertices of a graph by a subset of the vertices, using either the neighbourhood within the solution set or the distances to the solution vertices. Using a general reduction for this class of problems, we prove that the decision problems associated to these four notions are NP-complete, even for interval graphs of diameter and permutation graphs of diameter . While IDENTIFYING CODE and (OPEN) LOCATING-DOMINATING SET are trivially fixed-parameter-tractable when parameterized by solution size, it is known that in the same setting METRIC DIMENSION is -hard. We show that for interval graphs, this parameterization of METRIC DIMENSION is fixed-parameter-tractable.
Full work available at URL: https://arxiv.org/abs/1405.2424
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