Enumeration of Minimal Tropical Connected Sets

Abstract

A subset of vertices in a vertex-colored graph is called tropical if vertices of each color present in the subset. This paper is dedicated to the enumeration of all minimal tropical connected sets in various classes of graphs. We show that all minimal tropical connected sets can be enumerated in O( 1.7142 n ) time on n-vertex interval graph which improves previous O(1.8613 n ) upper bound obtained by Kratsch et al. Moreover, for chordal and general class of graphs we present algorithms with running times in O(1.937 n ) and O(1.999958 n ), respectively. The last two algorithms answer question implicitly asked in the paper [Kratsch et al. SOFSEM 2017]: «Is the number of tropical sets significantly smaller than the trivial upper bound 2 n ?».