# Bounds on global secure sets in cactus trees

Open Mathematics (2012)

- Volume: 10, Issue: 3, page 1113-1124
- ISSN: 2391-5455

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topKatarzyna Jesse-Józefczyk. "Bounds on global secure sets in cactus trees." Open Mathematics 10.3 (2012): 1113-1124. <http://eudml.org/doc/269551>.

@article{KatarzynaJesse2012,

abstract = {Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.},

author = {Katarzyna Jesse-Józefczyk},

journal = {Open Mathematics},

keywords = {Graph; Alliance; Secure set; Dominating set; Cactus tree; graph; alliance; secure set; dominating set; cactus tree},

language = {eng},

number = {3},

pages = {1113-1124},

title = {Bounds on global secure sets in cactus trees},

url = {http://eudml.org/doc/269551},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Katarzyna Jesse-Józefczyk

TI - Bounds on global secure sets in cactus trees

JO - Open Mathematics

PY - 2012

VL - 10

IS - 3

SP - 1113

EP - 1124

AB - Let G = (V, E) be a graph. A global secure set SD ⊆ V is a dominating set which satisfies the condition: for all X ⊆ SD, |N[X] ∩ SD| ≥ | N[X] − SD|. A global defensive alliance is a set of vertices A that is dominating and satisfies a weakened condition: for all x ∈ A, |N[x] ∩ A| ≥ |N[x] − A|. We give an upper bound on the cardinality of minimum global secure sets in cactus trees. We also present some results for trees, and we relate them to the known bounds on the minimum cardinality of global defensive alliances.

LA - eng

KW - Graph; Alliance; Secure set; Dominating set; Cactus tree; graph; alliance; secure set; dominating set; cactus tree

UR - http://eudml.org/doc/269551

ER -

## References

top- [1] Brigham R.C., Dutton R.D., Hedetniemi S.T., Security in graphs, Discrete Appl. Math., 2007, 155(13), 1708–1714 http://dx.doi.org/10.1016/j.dam.2007.03.009 Zbl1125.05071
- [2] Cami A., Balakrishnan H., Deo N., Dutton R.D., On the complexity of finding optimal global alliances, J. Combin. Math. Combin. Comput., 2006, 58, 23–31 Zbl1113.05074
- [3] Chellali M., Haynes T.W., Global alliances and independence in trees, Discuss. Math. Graph Theory, 2007, 27(1), 27, 19–27 http://dx.doi.org/10.7151/dmgt.1340 Zbl1189.05123
- [4] Eroh L., Gera R., Global alliance partition in trees, J. Combin. Math. Combin. Comput., 2008, 66, 161–169 Zbl1160.05022
- [5] Haynes T.W., Hedetniemi S.T., Henning M.A., Global defensive alliances in graphs, Electron. J. Combin., 2003, 10(1), #R47 Zbl1031.05096
- [6] Kristiansen P., Hedetniemi S.M., Hedetniemi S.T., Alliances in graphs, J. Combin. Math. Combin. Comput., 2004, 48, 157–177 Zbl1051.05068

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