Secure domination and secure total domination in graphs

A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V-X, there exists x ∈ X adjacent to u such that ${\displaystyle (X-\left\{x\right\})\cup \left\{u\right\}}$ is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number ${\displaystyle {\gamma }_s\left(G\right)\left({\gamma }_{st}\left(G\right)\right)}$. We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then ${\displaystyle {\gamma }_{st}\left(G\right)\le {\gamma }_s\left(G\right)}$. We also show that ${\displaystyle {\gamma }_{st}\left(G\right)}$ is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.