Tropical Dominating Sets in Vertex-Coloured Graphs
J.-A. Angles d'Auriac, Cs. Bujtas, A. El Maftouhi, M. Karpinski, Y., Manoussakis, L. Montero, N. Narayanan, L. Rosaz, J. Thapper, Zs. Tuza
TL;DR
This paper investigates the computational complexity and approximation strategies for finding minimum tropical dominating sets in vertex-coloured graphs, providing new bounds, algorithms, and hardness results.
Contribution
It introduces the NP-completeness of the problem on paths, derives upper bounds based on graph parameters, and develops an FPT algorithm for interval graphs.
Findings
NP-complete even on simple paths
Upper bounds related to degree and edges
FPT algorithm for interval graphs
Abstract
Given a vertex-coloured graph, a dominating set is said to be tropical if every colour of the graph appears at least once in the set. Here, we study minimum tropical dominating sets from structural and algorithmic points of view. First, we prove that the tropical dominating set problem is NP-complete even when restricted to a simple path. Then, we establish upper bounds related to various parameters of the graph such as minimum degree and number of edges. We also give upper bounds for random graphs. Last, we give approximability and inapproximability results for general and restricted classes of graphs, and establish a FPT algorithm for interval graphs.
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