Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity

Florent Foucaud (1) , George B. Mertzios (2) , Reza Naserasr (3) , Aline Parreau (4, 5) , Petru Valicov (6, 7)

1 LIMOS - Laboratoire d'Informatique, de Modélisation et d'optimisation des Systèmes
2 School of Engineering and Computing Sciences
3 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
4 LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
5 GOAL - Graphes, AlgOrithmes et AppLications
6 LIF - Laboratoire d'informatique Fondamentale de Marseille
7 ACRO - Algorithmique, Combinatoire et Recherche Opérationnelle

We consider the problems of finding optimal identifying codes, (open) locating-dominating sets and resolving sets (denoted Identifying Code, (Open) 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 2 and permutation graphs of diameter 2. 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 W[2]-hard. We show that for interval graphs, this parameterization of Metric Dimension is fixed-parameter-tractable.

Domaines: Mathématique discrète [cs.DM]

Florent Foucaud, George B. Mertzios, Reza Naserasr, Aline Parreau, Petru Valicov. Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity. Algorithmica, 2017, 78 (3), pp.914-944. ⟨10.1007/s00453-016-0184-1⟩. ⟨hal-01198784⟩