Annals of Communications in Mathematics

ABSTRACT. A dominating set \( S \) in \( G \) is called a pendant dominating set if \( \langle S \rangle \) contains at least one pendant vertex. The minimum cardinality of a pendant dominating set is called the pendant domination number, denoted by \( \gamma_{pe}(G) \). The pendant domination polynomial of \( G \) is denoted by \( D_{pe}(G,x) \) and is defined as\[D_{pe}(G,x)=\sum_{i=\gamma_{pe}(G)}^{n} d_{pe}(G,i)\,x^{i},\]where \( d_{pe}(G,i)x^{i} \) is the number of pendant dominating sets of size \( i \). In this paper, we obtained the pendant domination number and pendant domination polynomial of the corona of some graphs, namely, \( P_m \circ K_n \), \( C_m \circ K_n \), and \( K_m \circ K_n \).

Abstract: Let \( W_M \) be the wheel graph on \( M \geq 4 \) vertices and let \( \overline{K_n} \) be the independent graph on \( n \geq 1 \) vertices. We study the corona product \( W_M \circ K_n \) and obtain an explicit formula for its pendant domination polynomial. The computation starts from the domination polynomial and subtracts a correction term that counts dominating sets whose induced subgraph contains no vertex of degree 1. For the wheel, the correction term reduces to counting subsets of the rim cycle for which the selected rim vertices are not isolated on the rim. We also determine the pendant domination number for this family.