Happy Set Problem on Subclasses of Co-comparability Graphs

• Published in: Algorithmica Vol. 85; no. 11; pp. 3327 - 3347
• Main Authors: Eto, Hiroshi, Ito, Takehiro, Miyano, Eiji, Suzuki, Akira, Tamura, Yuma
• Format: Journal Article
• Language: English
• Published: New York Springer US 2023; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Apexes; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graphs; Mathematics of Computing; Permutations; Theory of Computation; Interval graphs; Permutation graphs; trapezoid graphs; Graph algorithm; Co-comparability graphs; Happy set problem
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-022-01081-0

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 3 n ) ; this improves the best known running time O ( k n 8 ) for interval graphs. We also design algorithms for n -vertex permutation graphs and d -trapezoid graphs which run in O ( n 2 + k 3 n ) and O ( n 2 + d 2 ( k + 1 ) 3 d n ) 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.