Table of Content
Samuel John E. Parreno
Open AccessArticlePendant Domination Polynomial of the Corona of a Wheel and an Independent Graph
Samuel John E. Parreno
Annals of Communications in Mathematics 2026,
(2026),
In Press
Abstract. Let WM be the wheel graph on M ≥ 4 vertices and let Kn be the independent graph on n ≥ 1 vertices. We study the corona product WM ◦ Kn 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.
Open AccessArticlet-Secure Hop Dominating Sets in Graphs
Samuel John E. Parreno
Annals of Communications in Mathematics ,
(2026),
in Press
Abstract. Hop domination was introduced as a distance-two analogue of domination and has been studied extensively in recent years. A secure hop dominating set, recently introduced, models a single adversarial attack at an unoccupied vertex (a vertex not in the current guard set) that can be defended by relocating one guard at distance two while preserving hop domination. Motivated by finite-order (multi-step) protection in classical secure domination, we introduce t-secure hop dominating sets (t ∈ N0), in which an adversary may launch a sequence of at most t attacks, each at a currently unoccupied vertex, and the defender responds by sequentially relocating one guard at distance two after each attack while maintaining hop domination throughout. Our main contribution is an exact correspondence: t-secure hop domination in a graph G is equivalent to smart t-secure domination in the hop graph H(G). This yields structural properties (monotonicity and additivity over components) and exact values for several graph families, including complete multipartite graphs, stars, paths, and cycles. In particular, we obtain closed formulas for γsh,t(Pn) and γsh,t(Cn) for all t ∈ N0, with explicit small-n exceptions in the cycle case.