The thesis first introduces the terminology and the theoretical basics necessary
to understand the concepts of security, domination, and secure domination in graphs.
In the second part of the thesis, graphs of Sierpiński will be defined. It will be
explained how they were created and what their characteristics are. To make it more
understandable, some small examples of Sierpiński graphs will also be drawn.
The proof of the theorem for the security number of Sierpiński graphs will be explained
and the necessary lemmas will be proved. We will also look for the secure domination
number of Sierpiński graphs. This problem will be divided into two parts, namely for the graphs
S_p^n with even p and for graphs S_p^n with odd p. For even p we will find the exact formula
for the secure domination number, for odd p we will only give the upper bound since finding
the exact formula is still an open problem.
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