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) Dominating ($f$-BDS). Let $G=(V,E)$ be a graph with color assignment function $\gamma \colon V \rightarrow \{1,\ldots,k\}$ that maps all vertices in $G$ onto $k$ colors. A subset $S\subseteq V$ is called $f$-balanced if $S$ contains $f$ from each class. In the $f$-BIS $f$-BDS problems, objective to compute an or $f$-balanced. show both are NP-complete even proper graphs. For problem, we design FPT algorithms, one parameterized by $(f,k)$ for graphs other vertex cover number general Moreover, optimization variant BIS simple greedy approach achieves approximation ratio $2$.