Minimum Consistent Subset in Interval Graphs and Circle Graphs

In a connected simple graph G = (V,E), each vertex of V is colored by a color from the set of colors C={c1, c2,..., c_{\alpha}}$. We take a subset S of V, such that for every vertex v in V\S, at least one vertex of the same color is present in its set of nearest neighbors in S. We refer to such a S as a consistent subset. The Minimum Consistent Subset (MCS) problem is the computation of a consistent subset of the minimum size. It is established that MCS is NP-complete for general graphs, including planar graphs. We expand our study to interval graphs and circle graphs in an attempt to gain a complete understanding of the computational complexity of the \mcs problem across various graph classes. This work introduces an (4\alpha+ 2)- approximation algorithm for MCS in interval graphs where \alpha is the number of colors in the interval graphs. Later, we show that in circle graphs, MCS is APX-hard.