Journal of Combinatorial Optimization

, Volume 35, Issue 4, pp 1312–1329 | Cite as

On the algorithmic aspects of strong subcoloring

  • M. A. Shalu
  • S. Vijayakumar
  • S. Devi Yamini
  • T. P. Sandhya
Article

Abstract

A partition of the vertex set V(G) of a graph G into \(V(G)=V_1\cup V_2\cup \cdots \cup V_k\) is called a k-strong subcoloring if \(d(x,y)\ne 2\) in G for every \(x,y\in V_i\), \(1\le i \le k\) where d(xy) denotes the length of a shortest x-y path in G. The strong subchromatic number is defined as \(\chi _{sc}(G)=\text {min}\{ k:G \text { admits a }k\)-\(\text {strong subcoloring}\}\). In this paper, we explore the complexity status of the StrongSubcoloring problem: for a given graph G and a positive integer k, StrongSubcoloring is to decide whether G admits a k-strong subcoloring. We prove that StrongSubcoloring is NP-complete for subcubic bipartite graphs and the problem is polynomial time solvable for trees. In addition, we prove the following dichotomy results: (i) for the class of \(K_{1,r}\)-free split graphs, StrongSubcoloring is in P when \(r\le 3\) and NP-complete when \(r>3\) and (ii) for the class of H-free graphs, StrongSubcoloring is polynomial time solvable only if H is an induced subgraph of \(P_4\); otherwise the problem is NP-complete. Next, we consider a lower bound on the strong subchromatic number. A strong set is a set S of vertices of a graph G such that for every \(x,y\in S\), \(d(x,y)= 2\) in G and the cardinality of a maximum strong set in G is denoted by \(\alpha _{s}(G)\). Clearly, \(\alpha _{s}(G)\le \chi _{sc}(G)\). We consider the complexity status of the StrongSet problem: given a graph G and a positive integer k, StrongSet asks whether G contains a strong set of cardinality k. We prove that StrongSet is NP-complete for (i) bipartite graphs and (ii) \(K_{1,4}\)-free split graphs, and it is polynomial time solvable for (i) trees and (ii) \(P_4\)-free graphs.

Keywords

Coloring Subcoloring Strong subcoloring Strong set 

Notes

Acknowledgements

The authors wish to thank the anonymous referees for their valuable comments on the content and presentation of this paper.

References

  1. Albertson MO, Jamison RE, Hedetniemi ST, Locke SC (1989) The subchromatic number of a graph. Discrete Math 74:33–49MathSciNetCrossRefMATHGoogle Scholar
  2. Bagan G, Merouane HB, Haddad M, Kheddouci H (2017) On some domination colorings of graphs. Discrete Appl Math 230:34–50MathSciNetCrossRefMATHGoogle Scholar
  3. Brooks RL (1941) On colouring the nodes of a network. Math Proc Camb Philos Soc 37:194197MathSciNetCrossRefGoogle Scholar
  4. Broersma H, Fomin FV, Nesetril J, Woeginger G (2002) More about subcolorings. Computing 69:187–203MathSciNetCrossRefMATHGoogle Scholar
  5. Fiala J, Jansen K, Le VB, Seidel E (2001) Graph subcolorings: complexity and algorithms. In: Brandst\(\ddot{\text{a}}\)dt A, Le VB (eds) Graph-theoretic concepts in computer science. WG 2001. Lecture notes in computer science, vol 2204. Springer, BerlinGoogle Scholar
  6. Fiala J, Jansen K, Le VB, Seidel E (2003) Graph subcolorings: complexity and algorithms. SIAM J Discrete Math 16:635–650MathSciNetCrossRefMATHGoogle Scholar
  7. Gandhi R, Greening B Jr, Pemmaraju S, Raman R (2010) Sub-coloring and hypo-coloring of interval graphs. Discrete Math Algorithms Appl 2:331–345MathSciNetCrossRefMATHGoogle Scholar
  8. Garey MR, Johnson DS (1990) Computers and intractability; a guide to the theory of NP-completeness. W. H. Freeman & Co., New YorkMATHGoogle Scholar
  9. Gavril F (1972) Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J Comput 1:180–187MathSciNetCrossRefMATHGoogle Scholar
  10. Holyer I (1981) The NP-completeness of edge-colouring. SIAM J Comput 10:718–720MathSciNetCrossRefMATHGoogle Scholar
  11. Illuri M, Renjith P, Sadagopan N (2016) Complexity of steiner tree in split graphs—dichotomy results. In: Algorithms and discrete applied mathematics, LNCS, vol 9602, pp 308–325Google Scholar
  12. Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum Press, New YorkGoogle Scholar
  13. Kral D, Kratochvil J, Tuza Z, Woeginger GJ (2001) Complexity of coloring graphs without forbidden induced subgraphs. In: Proceedings of the 27th international workshop on graph-theoretic concepts in computer science WG’01, LNCS, vol 2204, pp 254–262Google Scholar
  14. Krithika R, Rai A, Saurabh S, Tale P (2017) Parameterized and exact algorithms for class domination coloring. In: Proceedings of the 43\(^{{\rm rd}}\) conference on current trends in theory and practice of computer science (SOFSEM), LNCS, vol 10139, pp 336–349Google Scholar
  15. Merouane HB, Haddad M, Chellali M, Kheddouci H (2015) Dominated colorings of graphs. Graphs Comb 31:713–727MathSciNetCrossRefMATHGoogle Scholar
  16. Minty GJ (1980) On maximal independent sets of vertices in claw-free graphs. J Comb Theory B 28:284–304MathSciNetCrossRefMATHGoogle Scholar
  17. Shalu MA, Sandhya TP (2016) The cd-coloring of Graphs. In: Proceedings of the second international conference on algorithms and discrete applied mathematics (CALDAM), LNCS, vol 9602, pp 337–348Google Scholar
  18. Shalu MA, Vijayakumar S, Sandhya TP (2017) A lower bound of the cd-chromatic number and its complexity. In: Proceedings of the third international conference on algorithms and discrete applied mathematics (CALDAM), LNCS, vol 10156, pp 344–355Google Scholar
  19. Stacho J (2008) Complexity of generalized colourings of chordal graphs. Ph.D. Thesis, Simon Fraser UniversityGoogle Scholar
  20. Venkatakrishnan YB, Swaminathan V (2014) Color class domination numbers of some classes of graphs. Algebra Discrete Math 18:301–305MathSciNetMATHGoogle Scholar
  21. Vizing VG (1964) On an estimate of the chromatic class of a p-graph. Diskret Analiz 3:25–30 (Russian)MathSciNetGoogle Scholar
  22. West DB (2000) Introduction to graph theory, 2nd edn. Prentice-Hall, Upper Saddle RiverGoogle Scholar

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Authors and Affiliations

  1. 1.Indian Institute of Information Technology, Design and Manufacturing (IIITD&M)Kancheepuram, ChennaiIndia
  2. 2.VIT UniversityChennaiIndia
  3. 3.Department of ComputingThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong

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