# Balanced Independent and Dominating Sets on Colored Interval Graphs

We study two new versions of independent and dominating set problems on vertex-colored interval graphs, namely f-Balanced Independent Set (f-BIS) and f-Balanced Dominating Set (f-BDS). Let G=(V,E) be a vertex-colored interval graph with a k-coloring γ V →{1,...,k} for some k ∈N. A subset of vertices S⊆ V is called f-balanced if S contains f vertices from each color class. In the f-BIS and f-BDS problems, the objective is to compute an independent set or a dominating set that is f-balanced. We show that both problems are -complete even on proper interval graphs. For the BIS problem on interval graphs, we design two algorithms, one parameterized by (f,k) and the other by the vertex cover number of G. Moreover, we present a 2-approximation algorithm for a slight variation of BIS on proper interval graphs.

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