# Finding and counting small induced subgraphs efficiently

## Abstract

We give two algorithms for listing all simplicial vertices of a graph. The first of these algorithms takes *O*(*n*^{α}) time, where *n* is the number of vertices in the graph and *O*(*n*^{α}) 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 *K*_{4}. 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 *K*_{4}'s in a graph can be done within the same time bound \(O(e^{\tfrac{{\alpha + 1}}{2}} )\).

Using the result on the *K*_{4}'s we can count the number of occurences as induced subgraph of any other fixed connected graph on four vertices within *O*(*n*^{α}+e^{1.69}).

## Keywords

Adjacency Matrix Connected Graph Time Algorithm Recognition Algorithm Maximum Clique## Preview

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## References

- 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.Chiba, N. and T. Nishizeki, Arboricity and subgraph listing algorithms,
*SIAM J. Comput.*,**14**, (1985), pp. 210–223.Google Scholar - 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.Faudree, R., E. Flandrin and Z. Ryjáček, Claw-free graphs-A survey. Manuscript.Google Scholar
- 5.Harary, F.,
*Graph Theory*, Addison Wesley, Publ. Comp., Reading, Massachusetts, (1969).Google Scholar - 6.Itai, A. and M. Rodeh, Finding a minimum circuit in a graph,
*SIAM J. Comput.*,**7**, (1978), pp. 413–423.Google Scholar - 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.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.
- 10.Turán, P., Eine Extremalaufgabe aus der Graphentheorie,
*Mat. Fiz. Lapok*,**48**, (1941), pp. 436–452.Google Scholar - 11.Tucker, A., Coloring perfect
*K*_{4}— e-free graphs,*Journal of Combinatorial Theory, series B*,**42**, (1987), pp. 313–318.Google Scholar