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 $\mathcal{O}(n{k}^2)$. 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. Löffler, 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.

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References in corpus (1)

• An O(m) Algorithm for Cores Decomposition of Networks