Efficient enumeration of maximal k-degenerate induced subgraphs of a chordal graph. Conte, A., Kanté, M. M., Otachi, Y., Uno, T., & Wasa, K. *Theoretical Computer Science*, 2018. Paper doi abstract bibtex In this paper we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay O(m⋅ω(G)) for any n-vertex chordal graph with m edges, where ω(G)≤n is the maximum size of a clique in G. Degeneracy is a well known sparsity measure, and k-degenerate subgraphs are a notion of sparse subgraphs, which generalizes other problems such as independent sets (0-degenerate subgraphs) and forests (1-degenerate subgraphs). Many efficient enumeration algorithms are designed by solving the so-called Extension problem, which asks whether there exists a maximal solution containing a given set of nodes, but no node from a forbidden set. We show that solving this problem is np-complete for maximal k-degenerate induced subgraphs, motivating the need for additional techniques.

@article{Conte:Kante:Otachi:Uno:Wasa:TCS2018,
title = "Efficient enumeration of maximal k-degenerate induced subgraphs of a chordal graph",
journal = "Theoretical Computer Science",
year = "2018",
issn = "0304-3975",
doi = "10.1016/j.tcs.2018.08.009",
url = "http://www.sciencedirect.com/science/article/pii/S0304397518305255",
author = "Alessio Conte and Mamadou Moustapha Kanté and Yota Otachi and Takeaki Uno and Kunihiro Wasa",
keywords = "Graph enumeration, Graph algorithms, Polynomial delay, Degeneracy, Chordal graphs, Clique trees",
abstract = "In this paper we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay O(m⋅ω(G)) for any n-vertex chordal graph with m edges, where ω(G)≤n is the maximum size of a clique in G. Degeneracy is a well known sparsity measure, and k-degenerate subgraphs are a notion of sparse subgraphs, which generalizes other problems such as independent sets (0-degenerate subgraphs) and forests (1-degenerate subgraphs). Many efficient enumeration algorithms are designed by solving the so-called Extension problem, which asks whether there exists a maximal solution containing a given set of nodes, but no node from a forbidden set. We show that solving this problem is np-complete for maximal k-degenerate induced subgraphs, motivating the need for additional techniques."
}

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