Happy Set Problem on Subclasses of Co-comparability Graphs

In this paper, we investigate the complexity of the Maximum Happy Set problem on subclasses of co-comparability graphs. For a graph G and its vertex subset S, a vertex v∈ S is happy if all v’s neighbors in G are contained in S. Given a graph G and a non-negative integer k, Maximum Happy Set is the problem of finding a vertex subset S of G such that | S| = k and the number of happy vertices in S is maximized. In this paper, we first show that Maximum Happy Set is NP-hard even for co-bipartite graphs. We then give an algorithm for n-vertex interval graphs whose running time is O(n²+ k³n) ; this improves the best known running time O(kn⁸) for interval graphs. We also design algorithms for n-vertex permutation graphs and d-trapezoid graphs which run in O(n²+ k³n) and O(n²+ d²(k+ 1) ^3dn) time, respectively. These algorithmic results provide a nice contrast to the fact that Maximum Happy Set remains NP-hard for chordal graphs, comparability graphs, and co-comparability graphs.