# Distribution of global defensive *k*-alliances over some graph products

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## Abstract

If \(G=(V_G, E_G)\) is a graph, then \(S\subseteq V_G\) is a global defensive *k*-alliance in *G* if (i) each vertex not in *S* has a neighbor in *S* and (ii) each vertex of *S* has at least *k* more neighbors inside *S* than outside of it. The global defensive *k*-alliance number of *G* is the minimum cardinality among all global defensive *k*-alliance in *G*. In this paper this concept is studied on the generalized hierarchical, the lexicographic, the corona, and the edge corona product. For all of these products upper bounds expressed with related invariants of the factors are given. Sharpness of the bounds is also discussed.

## Keywords

Global alliance Global defensive*k*-alliance Hierarchical product of graphs Lexicographic product of graphs

## Mathematics Subject Classification

05C69 05C76## Notes

### Acknowledgements

Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (research core funding No. P1-0297 and projects J1-7110, J1-9109, N1-0043). The authors are indebted to the referees for helpful remarks which leaded us to correct and improve the paper.

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