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)


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).


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.


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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

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