Advertisement

# Finding and counting small induced subgraphs efficiently

• T. Kloks
• D. Kratsch
• H. Müller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1017)

## Abstract

We give two algorithms for listing all simplicial vertices of a graph. The first of these algorithms takes O(nα) time, where n is the number of vertices in the graph and O(nα) is the time needed to perform a fast matrix multiplication. The second algorithm can be implemented to run in $$O(e^{\tfrac{{2\alpha }}{{\alpha + 1}}} ) = O(e^{1.41} )$$, where e is the number of edges in the graph.

We present a new algorithm for the recognition of diamond-free graphs that can be implemented to run in time $$O(n^\alpha + e^{{3 \mathord{\left/{\vphantom {3 2}} \right.\kern-\nulldelimiterspace} 2}} )$$.

We also present a new recognition algorithm for claw-free graphs. This algorithm can be implemented to run in time $$O(e^{\tfrac{{\alpha + 1}}{2}} ) = O(e^{1.69} )$$.

It is a fairly easy observation that, within time $$O(e^{\tfrac{{\alpha + 1}}{2}} ) = O(e^{1.69} )$$ it can be checked whether a graph has a K4. This improves the $$O(e^{\tfrac{{3\alpha + 3}}{{\alpha + 3}}} ) = O(e^{1.89} )$$ algorithm mentioned by Alon, Yuster and Zwick.

Furthermore, we show that counting the number of K4's in a graph can be done within the same time bound $$O(e^{\tfrac{{\alpha + 1}}{2}} )$$.

Using the result on the K4's we can count the number of occurences as induced subgraph of any other fixed connected graph on four vertices within O(nα+e1.69).

## Keywords

Adjacency Matrix Connected Graph Time Algorithm Recognition Algorithm Maximum Clique
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Alon, N., R. Yuster and U. Zwick, Finding and counting given length cycles, Algorithms-ESA'94. Second Annual European Symposium, Springer-Verlag, Lecture Notes in Computer Science 855, (1994), pp. 354–364.Google Scholar
2. 2.
Chiba, N. and T. Nishizeki, Arboricity and subgraph listing algorithms, SIAM J. Comput., 14, (1985), pp. 210–223.Google Scholar
3. 3.
Corneil, D. G., Y. Perl and L. K. Stewart, A linear recognition algorithm for cographs, SIAM J. Comput., 4, (1985), pp. 926–934.Google Scholar
4. 4.
Faudree, R., E. Flandrin and Z. Ryjáček, Claw-free graphs-A survey. Manuscript.Google Scholar
5. 5.
Harary, F., Graph Theory, Addison Wesley, Publ. Comp., Reading, Massachusetts, (1969).Google Scholar
6. 6.
Itai, A. and M. Rodeh, Finding a minimum circuit in a graph, SIAM J. Comput., 7, (1978), pp. 413–423.Google Scholar
7. 7.
Minty, G. J., On maximal independent sets of vertices in claw-free graphs, J. Combin. Theory B, 28, (1980), pp. 284–304.Google Scholar
8. 8.
NešetŘil, J. and S. Poljak, On the complexity of the subgraph problem, Commentationes Mathematicae Universitatis Carolinae, 14 (1985), no. 2, pp. 415–419.Google Scholar
9. 9.
Olariu, S., Paw-free graphs, Information Processing Letters, 28, (1988), pp. 53–54.Google Scholar
10. 10.
Turán, P., Eine Extremalaufgabe aus der Graphentheorie, Mat. Fiz. Lapok, 48, (1941), pp. 436–452.Google Scholar
11. 11.
Tucker, A., Coloring perfect K 4 — e-free graphs, Journal of Combinatorial Theory, series B, 42, (1987), pp. 313–318.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1995

## Authors and Affiliations

• T. Kloks
• 1
• D. Kratsch
• 2
• H. Müller
• 2
1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyMB EindhovenThe Netherlands
2. 2.FakultÄt für Mathematik und InformatikFriedrich-Schiller-UniversitÄt JenaJenaGermany

## Personalised recommendations

### Citepaper 