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(n^2 + k^3n) ; this improves the best known running time O(kn^8) for interval graphs. We also design algorithms for n -vertex permutation graphs and d -trapezoid graphs which run in O(n^2 + k^3n) and O(n^2 + d^2(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.