Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds

Foucaud, F.; Mertzios, G.B.; Naserasr, R.; Parreau, A.; Valicov, P.

Abstract

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 find a subset of vertices, normally called a solution set, using which all vertices of the graph are distinguished. The identification can be done by considering the neighborhood within the solution set, or by employing the distances to the solution vertices. Normally the goal is to minimize the size of the solution set then. Here we study the case of interval graphs, unit interval graphs, (bipartite) permutation graphs and cographs. For these classes of graphs we give tight lower bounds for the size of such solution sets depending on the order of the input graph. While such lower bounds for the general class of graphs are in logarithmic order, the improved bounds in these special classes are of the order of either quadratic root or linear in terms of number of vertices. Moreover, the results for cographs lead to linear-time algorithms to solve the considered problems on inputs that are cographs.

Citation

Foucaud, F., Mertzios, G., Naserasr, R., Parreau, A., & Valicov, P. (2017). Identification, location–domination and metric dimension on interval and permutation graphs. I. Bounds. Theoretical Computer Science, 668, 43-58. https://doi.org/10.1016/j.tcs.2017.01.006

• Journal Article Type: Article
• Acceptance Date: Jan 4, 2017
• Online Publication Date: Jan 18, 2017
• Publication Date: Mar 15, 2017
• Deposit Date: Jan 21, 2017
• Publicly Available Date: Jan 18, 2018
• Journal: Theoretical Computer Science
• Print ISSN: 0304-3975
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 668
• Pages: 43-58
• DOI: https://doi.org/10.1016/j.tcs.2017.01.006
• Public URL: https://durham-repository.worktribe.com/output/1395853