# Protection of a network by complete secure domination

### Abstract

A complete secure dominating set of a graph G is a dominating set D \subseteq V(G) with the property that for each v \in D, there exists F={ v_{j} | v_{j} \in N(v) \cap (V(G)-D)} such that for each v_{j} \in F, ( D-{v }) \cup { v_{j} } is a dominating set. The minimum cardinality of any complete secure dominating set is called the complete secure domination number of G and is denoted by \gama_{csd}(G). In this paper, the bounds for complete secure domination number for some standard graphs like grid graphs and stacked prism graphs in terms of number of vertices of G are found and also the bounds for the complete secure domination number of a tree are obtained in terms of different parameters of G.

### References

V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).

S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).

N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).

P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.

J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.

E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.

W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).