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The cardinality of a largest independent set of G, denoted by α (G) , is called the independence number of G. The independent domination number i (G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by α c (G) , as the greatest integer r such that every vertex of G belongs to some independent subset X of V G with | X | ≥ r . The common independence number α c (G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality α c (G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i (G) ≤ α c (G) ≤ α (G) . In this paper, we characterize the trees T for which i (T) = α c (T) , and the block graphs G for which α c (G) = α (G) . [ABSTRACT FROM AUTHOR]

A total dominating set D of a graph G is said to be a secure total dominating set if for every vertex u ∈ V (G) \ D , there exists a vertex v ∈ D , which is adjacent to u, such that (D \ { v }) ∪ { u } is a total dominating set as well. The secure total domination number of G is the minimum cardinality among all secure total dominating sets of G. In this article, we obtain new relationships between the secure total domination number and other graph parameters: namely the independence number, the matching number and other domination parameters. Some of our results are tight bounds that improve some well-known results. [ABSTRACT FROM AUTHOR]