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Journal of Combinatorial Optimization

, Volume 30, Issue 3, pp 579–595 | Cite as

Progress on the Murty–Simon Conjecture on diameter-2 critical graphs: a survey

  • Teresa W. Haynes
  • Michael A. Henning
  • Lucas C. van der Merwe
  • Anders Yeo
Article

Abstract

A graph \(G\) is diameter \(2\)-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter-\(2\)-critical graph \(G\) of order \(n\) is at most \(\lfloor n^2/4 \rfloor \) and that the extremal graphs are the complete bipartite graphs \(K_{{\lfloor n/2 \rfloor },{\lceil n/2 \rceil }}\). We survey the progress made to date on this conjecture, concentrating mainly on recent results developed from associating the conjecture to an equivalent one involving total domination.

Keywords

Total domination Diameter-2-critical Total domination edge-critical 

Mathematical Subject Classification

05C69 

Notes

Acknowledgments

Michael A. Henning was supported in part by the South African National Research Foundation and a partial travel grant from East Tennessee State University and Anders Yeo was supported in part by East Tennessee State University

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Teresa W. Haynes
    • 1
    • 2
  • Michael A. Henning
    • 2
  • Lucas C. van der Merwe
    • 3
  • Anders Yeo
    • 2
  1. 1.Department of Mathematics and StatisticsEast Tennessee State UniversityJohnson CityUSA
  2. 2.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  3. 3.Department of MathematicsUniversity of Tennessee at ChattanoogaChattanoogaUSA

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