Metric Dimension Parameterized By Treewidth

Abstract A resolving set S of a graph G is subset its vertices such that no two have the same distance vector to . The Metric Dimension problem asks for minimum size, and in decision form, size at most some specified integer. This NP-complete, remains so very restricted classes graphs. It also W[2]-complete with respect solution. has proven elusive on graphs bounded treewidth. On algorithmic side, polynomial time algorithm known trees, even outerplanar graphs, but general case treewidth open. complexity parameterized hardness known. led several papers topic ask We provide first answer question. show by input W[1]-hard. More refinedly we prove that, unless Exponential Time Hypothesis fails, there solving $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o n -vertex constant degree, $$\text {pw}$$ pathwidth graph, f any computable function. stark contrast an FPT Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) combined parameter {tl}+\Delta$$ tl + Δ , where {tl}$$ tree-length $$\Delta$$ maximum-degree graph.