# A characterization of diameter-2-critical graphs with no antihole of length four

Open Mathematics (2012)

• Volume: 10, Issue: 3, page 1125-1132
• ISSN: 2391-5455

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## Abstract

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A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.

## How to cite

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Teresa Haynes, and Michael Henning. "A characterization of diameter-2-critical graphs with no antihole of length four." Open Mathematics 10.3 (2012): 1125-1132. <http://eudml.org/doc/269790>.

@article{TeresaHaynes2012,
abstract = {A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.},
author = {Teresa Haynes, Michael Henning},
journal = {Open Mathematics},
keywords = {Diameter critical; Diameter-2-critical; Antihole; Total domination critical; diameter critical; diameter-2-critical; antihole; total domination critical},
language = {eng},
number = {3},
pages = {1125-1132},
title = {A characterization of diameter-2-critical graphs with no antihole of length four},
url = {http://eudml.org/doc/269790},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Teresa Haynes
AU - Michael Henning
TI - A characterization of diameter-2-critical graphs with no antihole of length four
JO - Open Mathematics
PY - 2012
VL - 10
IS - 3
SP - 1125
EP - 1132
AB - A graph G is diameter-2-critical if its diameter is two and the deletion of any edge increases the diameter. In this paper we characterize the diameter-2-critical graphs with no antihole of length four, that is, the diameter-2-critical graphs whose complements have no induced 4-cycle. Murty and Simon conjectured that the number of edges in a diameter-2-critical graph of order n is at most n 2/4 and that the extremal graphs are complete bipartite graphs with equal size partite sets. As a consequence of our characterization, we prove the Murty-Simon Conjecture for graphs with no antihole of length four.
LA - eng
KW - Diameter critical; Diameter-2-critical; Antihole; Total domination critical; diameter critical; diameter-2-critical; antihole; total domination critical
UR - http://eudml.org/doc/269790
ER -

## References

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8. [8] Haynes T.W., Henning M.A., van der Merwe L.C., Yeo A., On a conjecture of Murty and Simon on diameter 2-critical graphs, Discrete Math., 2011, 311(17), 1918–1924 http://dx.doi.org/10.1016/j.disc.2011.05.007 Zbl1223.05050
9. [9] Haynes T.W., Henning M.A., Yeo A., A proof of a conjecture on diameter 2-critical graphs whose complements are claw-free, Discrete Optim., 2011, 8(3), 495–501 http://dx.doi.org/10.1016/j.disopt.2011.04.003 Zbl1236.05147
10. [10] Henning M.A., A survey of selected recent results on total domination in graphs, Discrete Math., 2009, 309(1), 32–63 http://dx.doi.org/10.1016/j.disc.2007.12.044 Zbl1219.05121
11. [11] van der Merwe L.C., Total Domination Critical Graphs, PhD thesis, University of South Africa, 1998 Zbl0918.05069
12. [12] van der Merwe L.C., Mynhardt C.M., Haynes T.W., Total domination edge critical graphs, Util. Math., 1998, 54, 229–240 Zbl0918.05069
13. [13] Murty U.S.R., On critical graphs of diameter 2, Math. Mag., 1968, 41, 138–140 http://dx.doi.org/10.2307/2688184 Zbl0167.22102
14. [14] Plesník J., Critical graphs of given diameter, Acta Fac. Rerum Natur. Univ. Comenian. Math., 1975, 30, 71–93 (in Slovak) Zbl0318.05115
15. [15] Xu J.M., Proof of a conjecture of Simon and Murty, J. Math. Res. Exposition, 1984, 4, 85–86 (in Chinese) Zbl0569.05031

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