Abstract:

A set \( S \subseteq V(G) \) of the connected graph \( G = (V(G), E(G)) \) is said to be a connected secure dominating set if \( S \) is a dominating set, \( S \) is a secure set, and \( \langle S \rangle_G \) is a connected graph. The connected secure domination polynomial of \( G \) is \( D_s^c(G,x) = \sum_{i=\gamma_s^c(G)}^{n} d_s^c(G,i)x^i \) where \( \gamma_s^c(G) = \min \{|S| : S \text{ is a connected secure dominating set of } G\} \) and \( d_s^c(G,i) \) is the number of connected secure dominating sets with cardinality \( i \). In this paper, we will determine the connected secure domination polynomial of path graph \( P_n \), cycle graph \( C_n \), complete graph \( K_n \), star graph \( K_{1,n} \), and corona graph \( G \circ K_1 \).