The (weighted) metric dimension of graphs: hard and easy cases

L. Epstein, A. Levin, G.J. Woeginger

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

11 Citations (Scopus)
93 Downloads (Pure)

Abstract

For an undirected graph G¿=¿(V,E), we say that for l,u,v¿¿¿V, l separates u from v if the distance between u and l differs from the distance from v to l. A set of vertices L¿¿¿V is a feasible solution if for every pair of vertices u,v¿¿¿V there is l¿¿¿L that separates u from v. The metric dimension of a graph is the minimum cardinality of such a feasible solution. Here, we extend this well-studied problem to the case where each vertex v has a non-negative cost, and the goal is to find a feasible solution with a minimum total cost. This weighted version is NP-hard since the unweighted variant is known to be NP-hard. We show polynomial time algorithms for the cases where G is a path, a tree, a cycle, a cograph, a k-edge-augmented tree (that is, a tree with additional k edges) for a constant value of k, and a (not necessarily complete) wheel. The results for paths, trees, cycles, and complete wheels extend known polynomial time algorithms for the unweighted version, whereas the other results are the first known polynomial time algorithms for these classes of graphs even for the unweighted version. Next, we extend the set of graph classes for which computing the unweighted metric dimension of a graph is known to be NPC by showing that for split graphs, bipartite graphs, co-bipartite graphs, and line graphs of bipartite graphs, the problem of computing the unweighted metric dimension of the graph is NPC.
Original languageEnglish
Title of host publicationGraph-Theoretic Concepts in Computer Science (38th International Workshop, WG 2012, Jerusalem, Israel, June 26-28, 2012, Revised Selected Papers)
EditorsM.C. Golumbic, M. Stern, A. Levy, G. Morgenstern
Place of PublicationBerlin
PublisherSpringer
Pages114-125
ISBN (Print)978-3-642-34610-1
DOIs
Publication statusPublished - 2012
Event38th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2012), June 26-28, 2012, Jerusalem, Israel - Jerusalem, Israel
Duration: 26 Jun 201228 Jun 2012
https://sites.google.com/site/wg2012conf/

Publication series

NameLecture Notes in Computer Science
Volume7551
ISSN (Print)0302-9743

Workshop

Workshop38th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2012), June 26-28, 2012, Jerusalem, Israel
Abbreviated titleWG 2012
CountryIsrael
CityJerusalem
Period26/06/1228/06/12
Internet address

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