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

Teresa W. Haynes (), Michael A. Henning, Lucas C. Merwe and Anders Yeo
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Teresa W. Haynes: East Tennessee State University
Michael A. Henning: University of Johannesburg
Lucas C. Merwe: University of Tennessee at Chattanooga
Anders Yeo: University of Johannesburg

Journal of Combinatorial Optimization, 2015, vol. 30, issue 3, No 11, 579-595

Abstract: Abstract A graph $$G$$ G is diameter $$2$$ 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- $$2$$ 2 -critical graph $$G$$ G of order $$n$$ n is at most $$\lfloor n^2/4 \rfloor $$ ⌊ n 2 / 4 ⌋ and that the extremal graphs are the complete bipartite graphs $$K_{{\lfloor n/2 \rfloor },{\lceil n/2 \rceil }}$$ K ⌊ n / 2 ⌋ , ⌈ n / 2 ⌉ . 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; 05C69 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10878-013-9651-7

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