(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this paper, we revisit the concept of secure hop domination in graphs and define a new concept called secure pointwise non-domination. A pointwise non-dominating set S is a secure pointwise non-dominating set if for every u V (G) S, there exists v S N_G(u) such that (S v) u is a pointwise non-dominating set. The secure pointwise non-domination number spnd(G) of G is the smallest cardinality of a secure pointwise non-dominating set in G. In this paper, we give bounds on the secure pointwise non-domination number and characterize those graphs which attain these bounds. We also determine the secure pointwise non-domination number of some classes of graphs. Necessary and sufficient conditions for a subset in the join of graphs to be a secure hop dominating set is given. Moreover, we show that given positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph such that γ_h(G) = a and γ_sh(G) = b, where γ_h(G) and γ_sh(G) are the hop domination number and secure hop domination number of G, respectively.