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

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 $2$ 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.

22 pages, 9 figures. Some theorems have been restated and errors have been corrected

References in corpus (2)

• Identifying codes in hereditary classes of graphs and VC-dimension
• Metric Dimension Parameterized by Max Leaf Number

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