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

Noga Mordechai Alon, Shai Gutner

Research output: Chapter in Book/Report/Conference proceedingConference contribution

11 Scopus citations

Abstract

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.

Original languageEnglish (US)
Title of host publicationComputing and Combinatorics - 13th Annual International Conference, COCOON 2007, Proceedings
Pages394-405
Number of pages12
StatePublished - Dec 1 2007
Externally publishedYes
Event13th Annual International Computing and Combinatorics Conference, COCOON 2007 - Banff, Canada
Duration: Jul 16 2007Jul 19 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4598 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other13th Annual International Computing and Combinatorics Conference, COCOON 2007
CountryCanada
CityBanff
Period7/16/077/19/07

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)

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    Alon, N. M., & Gutner, S. (2007). Linear time algorithms for finding a dominating set of fixed size in degenerated graphs. In Computing and Combinatorics - 13th Annual International Conference, COCOON 2007, Proceedings (pp. 394-405). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4598 LNCS).