On Disjoint Cycles

It is shown, that for each constant k ≥ 1, the following problems can be solved in O(n) time: given a graph G, determine whether G has k vertex disjoint cycles, determine whether G has k edge disjoint cycles, determine whether G has a feedback vertex set of size ≤ k. Also, every class \(\mathcal{G}\), that is closed under minor taking, or that is closed under immersion taking, and that does not contain the graph formed by taking the disjoint union of k copies of K3, has an \(\mathcal{O}\)(n) membership test algorithm.

[1]  Detlef Seese,et al.  Problems Easy for Tree-Decomposable Graphs (Extended Abstract) , 1988, ICALP.

[2]  Hans L. Bodlaender,et al.  On Linear Time Minor Tests and Depth First Search , 1989, WADS.

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  Michael R. Fellows,et al.  On search decision and the efficiency of polynomial-time algorithms , 1989, STOC '89.

[5]  Stefan Arnborg,et al.  Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey , 1985, BIT.

[6]  Bruno Courcelle,et al.  An algebraic theory of graph reduction , 1990, JACM.

[7]  Stephen T. Hedetniemi,et al.  Linear algorithms on k-terminal graphs , 1987 .

[8]  Hans L. Bodlaender,et al.  NC-Algorithms for Graphs with Small Treewidth , 1988, WG.

[9]  Michael R. Fellows,et al.  Polynomial-time self-reducibility: theoretical motivations and practical results ∗ , 1989 .

[10]  P. Seymour,et al.  Surveys in combinatorics 1985: Graph minors – a survey , 1985 .

[11]  Jan van Leeuwen,et al.  Graph Algorithms , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[12]  Hans L. Bodlaender,et al.  Some Classes of Graphs with Bounded Treewidth , 1988, Bull. EATCS.

[13]  Paul D. Seymour,et al.  Graph minors. VI. Disjoint paths across a disc , 1986, J. Comb. Theory, Ser. B.

[14]  J. Lagergren.  Efficient parallel algorithms for tree-decomposition and related problems , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.