Distinguishing problems in interval graphs

Aline Parreau
CNRS - Universit\'e Lyon 1 (France)

Florent Foucaud
Universit\'e Blaise Pascal - Clermont-Ferrand (France)

George Mertzios
Durham University (United Kingdom)

Reza Naserasr
CNRS - Universit\'e Paris Sud (France)

Petru Valicov
Universit\'e Aix-Marseille (France)



Content: We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets 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 graphs that are at the same time interval graphs and permutation graphs and have diameter 2. While finding an optimal identifying code and an (open) locating-dominating set are trivially fixed-parameter-tractable when parametrized by solution size, it is known that in the same setting finding the metric dimension of a graph (which is the size of an optimal resolving set) is W[2]-hard. We show that for interval graphs, this parametrization of metric dimension is fixed parameter tractable.

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