TY - GEN

T1 - Linear time algorithms for finding a dominating set of fixed size in degenerated graphs

AU - Alon, Noga

AU - Gutner, Shai

PY - 2007

Y1 - 2007

N2 - There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a kO(dk)n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain Kh as a topological minor, we give an improved algorithm for the problem with running time (O(h))hkn. For graphs which are Kh-minor-free, the running time is further reduced to (O(log h))hk/2n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(n log n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs.

AB - There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a kO(dk)n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain Kh as a topological minor, we give an improved algorithm for the problem with running time (O(h))hkn. For graphs which are Kh-minor-free, the running time is further reduced to (O(log h))hk/2n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(n log n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs.

KW - Degenerated graphs

KW - Dominating set problem

KW - Finding an induced cycle

KW - Fixed-parameter tractable algorithms

KW - H-minor-free graphs

UR - http://www.scopus.com/inward/record.url?scp=37849019999&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37849019999&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:37849019999

SN - 9783540735441

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 394

EP - 405

BT - Computing and Combinatorics - 13th Annual International Conference, COCOON 2007, Proceedings

T2 - 13th Annual International Computing and Combinatorics Conference, COCOON 2007

Y2 - 16 July 2007 through 19 July 2007

ER -