New algorithms for $k$-degenerate graphs

A graph is k-degenerate if any induced subgraph has a vertex of degree at most k. In this paper we prove new algorithms finding cliques and similar structures in these graphs. We design linear time Fixed-Parameter Tractable algorithms for induced and non induced bicliques. We prove an algorithm listing all maximal bicliques in time O(k(n−k)2), improving the result of [D. Eppstein, Arboricity and bipartite subgraph listing algorithms, Information Processing Letters, (1994)]. We construct an algorithm listing all cliques of size l in optimal time O(l(n − k) ( k l−1 ) ), improving a result of [N. Chiba and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM, (1985)]. As a consequence we can list all triangles in such graphs in optimal time O((n− k)k) improving the previous bound of O(nk). We show another optimal algorithm listing all maximal cliques in time O(k(n − k)3), matching the best possible complexity proved in [D. Eppstein, M. Lffler, and D. Strash, Listing all maximal cliques in large sparse real-world graphs, JEA, (2013)]. Finally we prove polynomial (2− 1 k ) and O(k(log log k) /(log k))-approximation algorithms for the minimum vertex cover and the maximum clique problems, respectively.