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. * A short version of this paper, containing only the results about location-domination and metric dimension, appeared in the proceedings of the WG 2015 conference [28].pair in X has a vertex in S (totally) separating it. We have the three key definitions, that merge the concepts of (total) domination and (total) separation:Definition 1 (Slater [52,53], Babai [3]). A set S of vertices of a graph G is a locating-dominating set if it is a dominating set and it separates the vertices of V (G) \ S.

Algorithms and Complexity for Metric Dimension and Location-domination on Interval and Permutation Graphs

“…However, we do not know their complexity for unit interval graphs or bipartite permutation graphs (note that both problems are polynomial-time solvable on chain graphs, a subclass of bipartite permutation graphs [12]). We also note that our reduction can be adapted to related problems such as Identifying Code (see the full version of this paper [16]). …”

“…We dene two natural total orderings of V (G) based on this model: x < L y if and only if the left endpoint of x is smaller then the left endpoint of y, and x < R y if and only if the right endpoint of x is smaller than the right endpoint of y. We will work with the fourth distance-power G 4 of the input graph G which is also an interval graph and has an interval model inducing the same orders < L and < R as G [17].…”

“…Since u has started before v but has not stopped before the start of v, both u, v belong to X t , proving Property (ii). is an interval graph, and it has an intersection model inducing the same orders < L and < R [17]). Then the following holds.…”

Identification of points using disks

“…If the points of P are colinear, then G is a unit interval graph. The complexity of identifying codes in unit interval graphs is surprisingly still open [9] (but has been proved to be NP-complete for interval graphs). Junnila and Laihonen [11] studied identifying codes in the grid Z 2 using Euclidean balls.…”

“…• P 2,9 can be identified by the set of disks : D [3,4] [ 1,6] , D [4,6] [ 2,9] , D [6,7] [ 4,8] , D [1,8] [ 3,4] and D [2,9] [ 6,7] .…”