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).

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Copyright information

© Springer-Verlag 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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