New algorithms for $k$-degenerate graphs

Abstract: A graph is $k$-degenerate if any induced subgraph has a vertex of degree at most $k$. In this paper we prove new algorithms for cliques and similar structures for 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 $\mathcal{O}(k^{3}(n-k)2^{k})$, 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 time $\mathcal{O}(l(n-k)k(k-1)^{l-2})$, 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 time $\mathcal{O}((n-k)k^{2})$ improving the previous bound of $\mathcal{O}(nk^2)$. We show another optimal algorithm listing all maximal cliques in time $\mathcal{O}(k(n-k)3^{k/3})$, matching the best possible complexity proved in [D. Eppstein, M. L\"offler, and D. Strash, Listing all maximal cliques in large sparse real-world graphs, JEA, (2013)]. Finally we prove $(2-\frac{1}{k})$ and $\mathcal{O}(k(\log\log k)^{2}\slash (\log k)^{3})$-approximation algorithms for the minimum vertex cover and the maximum clique problems, respectively.