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Polynomial time algorithm for k-vertex-edge dominating problem in interval graphs

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Abstract

Let G be a connected interval graph with n vertices and m edges. For any positive integer k and any subset S of E(G), we design an \(O(k|S|+m)\) time algorithm to find a minimum k-vertex-edge dominating set of G with respect to S. This shows that the vertex-edge domination problem and the double vertex-edge domination problem can be solved in linear time. Furthermore, the k-vertex-edge domination problem can also be solved in O(km) time algorithm in interval graphs. Finally, we present a linear time algorithm to solve the independent vertex-edge domination problem for unit interval graphs.

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References

  • Boutrig R, Chellali M, Haynes TW, Hedetniemi ST (2016) Vertex-edge domination in graphs. Aequ Math 90(2):355–366

    Article  MathSciNet  MATH  Google Scholar 

  • Boutrig R, Chellali M (2018) Total vertex-edge domination. Int J Comput Math 95(9):1820–1828

    Article  MathSciNet  MATH  Google Scholar 

  • Fishburn PC (1985) Interval orders and interval graphs: a study of partially ordered sets. Wiley

    Book  MATH  Google Scholar 

  • Golumbic MC (2004) Algorithmic graph theory and perfect graphs, vol 57, 2nd edn. Annals of Discrete Mathematics. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Haynes TW, Hedetniemi ST, Slater PJ (1998a) Domination in graphs: advanced topics. Marcel Dekker Inc, New York

  • Haynes TW, Hedetniemi ST, Slater PJ (1998b) Fundamentals of domination in graphs. Marcel Dekker Inc, New York

  • Krishnakumari B, Venkatakrishnan YB, Krzywkowski M (2014) Bounds on the vertex-edge domination number of a tree. C R Math 352(5):363–366

    Article  MathSciNet  MATH  Google Scholar 

  • Krishnakumari B, Chellali M, Venkatakrishnan YB (2017) Double vertex-edge domination. Discrete Math Algorithms Appl 09(04):1750045

    Article  MathSciNet  MATH  Google Scholar 

  • Lewis JR (2007) Vertex-edge and edge-vertex parameters in graphs. Ph.D thesis, Clemson

  • Lewis JR, Hedetniemi ST, Haynes TW, Fricke GH (2010) Vertex-edge domination. Util Math 81:193–213

    MathSciNet  MATH  Google Scholar 

  • Li P, Wang A (2022) Polynomial Time Algorithm for \(k\)-vertex-edge Dominating Problem in Interval Graphs. AAIM, LNCS 13513

  • Li P, Wu Y (2015) Spanning connectedness and Hamiltonian thickness of graphs and interval graphs. Discrete Math Theor Comput Sci 16(2):125–210

    Article  MathSciNet  MATH  Google Scholar 

  • Li P, Wu Y (2017) A linear time algorithm for the 1-fixed-endpoint path cover problem on interval graphs. SIAM J Discrete Math 31(1):210–239

    Article  MathSciNet  MATH  Google Scholar 

  • Möhring RH (1985) Algorithmic aspects of comparability graphs and interval graphs. In: Rival I (ed) Graphs and orders. D. Reidel, Boston, pp 41–101

    Chapter  Google Scholar 

  • Paul S, Ranjan K (2021) Results on vertex-edge and independent vertex-edge domination. J Comb Optim 44:303

    Article  MathSciNet  MATH  Google Scholar 

  • Peters JKW (1986) Theoretical and algorithmic results on domination and connectivity (Nordhaus-Gaddum, Gallai type results, max–min relationships, linear time, series-parallel). Ph.D thesis, Clemson

  • Ramalingam G, Rangan CP (1988) A uniform approach to domination problems on interval graphs. Inf Process Lett 27:271–274

    Article  MATH  Google Scholar 

  • Raychaudhuri A (1987) On powers of interval and unit interval graphs. Congr Numer 59:235–242

    MathSciNet  MATH  Google Scholar 

  • Shang J, Li P, Shi Y (2021) The longest cycle problem is polynomial on interval graphs. Theor Comput Sci 859:37–47

    Article  MathSciNet  MATH  Google Scholar 

  • Trotter WT (1997) New perspectives on interval orders and interval graphs. In: Bailey RA (ed) Surveys in combinatorics, vol 241. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, pp 237–286

    Google Scholar 

  • Zylinski P (2019) Vertex-edge domination in graphs. Aequ Math 93(4):735–742

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

We thank the referees and editors for their constructive input. This work was supported by the National Natural Science Foundation of China (11701059), the Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0272, cstc2021jcyj-msxmX0436), the Youth project of science and technology research program of Chongqing Education Commission of China (KJQN202 001130, KJQN202001107, KJQN202101130).

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  1. Chongqing University of Technology, 69 Hongguang Road, Chongqing, China

    Peng Li & Aifa Wang

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  1. Peng Li
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  2. Aifa Wang
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Correspondence to Aifa Wang.

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This is the updated version of a conference paper that was presented at AAIM 2022 (Li and Wang 2022).

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Li, P., Wang, A. Polynomial time algorithm for k-vertex-edge dominating problem in interval graphs. J Comb Optim 45, 45 (2023). https://doi.org/10.1007/s10878-022-00982-8

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