RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Prikl. Diskr. Mat. Suppl.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Prikl. Diskr. Mat. Suppl., 2014, Issue 7, Pages 154–157 (Mi pdma136)  

Computational methods in discrete mathematics

Structural decomposition of graphs and its application for solving optimization problems on sparse graphs

V. V. Bykova

Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk

Abstract: The concept of a sparse graph is formalized through a numeric parameter called the treewidth of the graph. This parameter characterizes the size of cliques and separators of the graph. Sparse graphs have small treewidth. A decomposition technique for solving optimization problems on sparse graphs is proposed. This technique implements the principle of “divide and rule” and is based on the atomic representation of the input graph. By an atom of a graph is meant a maximal connected subgraph having no a clique being a minimal separator. An algorithm that creates the atomic representation of the input graph is presented. It is shown that this algorithm for graph $G=(V,E)$ builds the atomic representation in time $\mathrm O(|V|^\tau)$, $2\leq\tau<3$. The atomic representation of the graph is a generalization of the decomposition of the graph into blocks using articulation points. If only clique minimum separators are used, then the set of atoms is unique to a given graph. The important properties of atoms are presented. Based on these properties, atomic representation can be used for optimization problems, which are based on the ratio of the adjacency of vertices and edges. For two problems, called All-Pairs Shortest-Path and Maximum-Clique-Problem, the results of the application of atomic representation are presented. It is shown that the time of solving these problems depends linearly on the number of vertices of the input graph. Of course, this is true if the input graph has bounded treewidth. The obtained results make it possible to handle sparse graphs, which have a very large number of vertices. For example, such graphs are Small World Graphs.

Keywords: sparse graph, graph algorithms, treewidth, tree decomposition, atom graph.

Full text: PDF file (414 kB)
References: PDF file   HTML file

Document Type: Article
UDC: 519.178

Citation: V. V. Bykova, “Structural decomposition of graphs and its application for solving optimization problems on sparse graphs”, Prikl. Diskr. Mat. Suppl., 2014, no. 7, 154–157

Citation in format AMSBIB
\Bibitem{Byk14}
\by V.~V.~Bykova
\paper Structural decomposition of graphs and its application for solving optimization problems on sparse graphs
\jour Prikl. Diskr. Mat. Suppl.
\yr 2014
\issue 7
\pages 154--157
\mathnet{http://mi.mathnet.ru/pdma136}


Linking options:
  • http://mi.mathnet.ru/eng/pdma136
  • http://mi.mathnet.ru/eng/pdma/y2014/i7/p154

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Prikladnaya Diskretnaya Matematika. Supplement
    Number of views:
    This page:198
    Full text:97
    References:24

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2018