On the Number of Maximal Independent Sets in a Graph
DOI:
https://doi.org/10.7146/brics.v9i15.21733Resumé
We show that the number of maximal independent sets of size exactly k in any graph of size n is at most [ n/k ]^{k-(n mod k)} ([ n/k ] +1)^{n mod k}. For maximal independent sets of size at most k the same bound holds for k <= n/3. For k > n/3 a bound of approximately 3^{n/3} is given. All the bounds are exactly tight and improve Eppstein (2001) who give the bound 3^{4k-n}4^{n-3k} on the number of maximal independent sets of size at most k, which is the same for n/4 <= k <= n/3, but larger otherwise. We give an algorithm listing the maximal independent sets in a graph in time proportional to these bounds (ignoring a polynomial factor), and we use this algorithm to construct algorithms for 4- and 5- colouring running in time O(1.7504^n) and O(2.1593^n), respectively.Downloads
Publiceret
2002-04-05
Citation/Eksport
Nielsen, J. M. (2002). On the Number of Maximal Independent Sets in a Graph. BRICS Report Series, 9(15). https://doi.org/10.7146/brics.v9i15.21733
Nummer
Sektion
Artikler
Licens
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).