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Distribution of global defensive k-alliances over some graph products

  • Mostafa TavakoliEmail author
  • Sandi Klavžar
Original Paper
  • 27 Downloads

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesFerdowsi University of MashhadMashhadIran
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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