Skip to main content
SpringerLink
Log in

NP-Completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

This paper considers the two-step colouring problem for an undirected connected graph. The problem is about colouring the graph in a given number of colours so that no pair of vertices at a distance of 1 or 2 between each other has the same colour. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, the polynomial reduction of the problems to one another is analyzed and proved. In particular, this allows us to prove the NP-completeness of the two-step colouring problem. Also we specify some of its properties. The two-step colouring problem as applied to rectangular grid graphs is considered separately. The maximum vertex degree in these graphs ranges from 0 to 4. The function of two-vertex colouring in the minimum possible number of colours was defined and proved for each case. The resulting functions are drawn so that each vertex is coloured independently from others. If the vertices are examined in a sequence, the time for colouring a rectangular grid graph will be polynomial.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.
Fig. 17.
Fig. 18.

Similar content being viewed by others

REFERENCES

  1. Korsakov, S.V., Smirnov, A.V., and Sokolov, V.A., Principles of organizing the interoperability of equipollent nodes in a wireless mesh-network with time division multiple access, Autom. Control Comput. Sci., 2016, vol. 50, no. 6, pp. 415–422.

    Article  Google Scholar 

  2. Harary, F., Graph Theory, Addison-Wesley Pub. Co., 1969.

    Book  Google Scholar 

  3. Garey, M.R. and Johnson, D.S., Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, 1979.

    MATH  Google Scholar 

  4. Rublev, V.S., Elementy teorii grafov. Izomorfizm, planarnost’, marshruty v grafakh (Elements of Graph Theory. Isomorphism, Planarity, Routes in Graphs), Yaroslavl: Yarosl. Gos. Univ., 2010.

  5. Umans, C. and Lenhart, W., Hamiltonian cycles in solid grid graphs, Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1997, pp. 496–505.

  6. Keshavarz-Kohjerdi, F. and Bagheri, A., An efficient parallel algorithm for the longest path problem in meshes, J. Supercomput., 2013, vol. 65, no. 2, pp. 723–741.

    Article  Google Scholar 

Download references

Funding

The study was funded by the Russian Foundation for Basic Research, scientific project no. 17-07-00823 A.

Author information

Authors and Affiliations

  1. Demidov Yaroslavl State University, 150003, Yaroslavl, Russia

    N. S. Medvedeva & A. V. Smirnov

Authors
  1. N. S. Medvedeva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. V. Smirnov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to N. S. Medvedeva or A. V. Smirnov.

Ethics declarations

CONFLICT OF INTEREST

The authors declare that they have no conflicts of interest.

ADDITIONAL INFORMATION

Natalya S. Medvedeva, orcid.org/0000-0002-1632-5411, Student.

Alexander V. Smirnov, orcid.org/0000-0002-0980-2507, PhD, Associate Professor.

Additional information

Translated by S. Kuznetsov

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Medvedeva, N.S., Smirnov, A.V. NP-Completeness and One Polynomial Subclass of the Two-Step Graph Colouring Problem. Aut. Control Comp. Sci. 54, 685–696 (2020). https://doi.org/10.3103/S0146411620070159

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411620070159

Keywords:

Search

Navigation