Tropical dominating sets in vertex-coloured 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 an optimal upper bound 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.

Enumeration of Minimal Tropical Connected Sets

“…There are papers dedicated to the study of various variants of tropical sets. For example tropical dominating sets were studied in [16], tropical matchings in [10], tropical paths in [9], tropical vertex-disjoint cycles in [31]. However, it seems that tropical connected sets attract the greatest attention [7,8,17,27].…”