Efficient enumeration of maximal k-degenerate induced subgraphs of a chordal graph

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.