Abstract
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 \(v \in S\) 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 \(O(k^2n^2)\); this improves the best known running time \(O(kn^8)\) for interval graphs. We also design an algorithm for n-vertex permutation graphs whose running time is \(O(k^3n^2)\). These two 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.
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Notes
- 1.
We note that the graph coloring problem introduced by Zhang and Li [11] is called a similar name, Maximum Happy Vertices, but it is a different problem from ours.
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Acknowledgments
This work is partially supported by JSPS KAKENHI Grant Numbers JP18H04091, JP19K11814, JP20H05793, JP20H05794, JP20K11666, JP21K11755 and JP21K21302, Japan.
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Eto, H., Ito, T., Miyano, E., Suzuki, A., Tamura, Y. (2022). Happy Set Problem on Subclasses of Co-comparability Graphs. In: Mutzel, P., Rahman, M.S., Slamin (eds) WALCOM: Algorithms and Computation. WALCOM 2022. Lecture Notes in Computer Science(), vol 13174. Springer, Cham. https://doi.org/10.1007/978-3-030-96731-4_13
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DOI: https://doi.org/10.1007/978-3-030-96731-4_13
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