TY - GEN

T1 - The (Weighted) metric dimension of graphs

T2 - 38th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2012

AU - Epstein, Leah

AU - Levin, Asaf

AU - Woeginger, Gerhard J.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84868035478&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-34611-8_14

DO - 10.1007/978-3-642-34611-8_14

M3 - Conference contribution

AN - SCOPUS:84868035478

SN - 9783642346101

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 114

EP - 125

BT - Graph-Theoretic Concepts in Computer Science - 38th International Workshop, WG 2012, Revised Selcted Papers

Y2 - 26 June 2012 through 28 June 2012

ER -