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Let G = (V , E) be a graph, and let β ∈ R . Motivated by a service coverage maximization problem with limited resources, we study the β -differential of G. The β-differential of G, denoted by ∂ β (G) , is defined as ∂ β (G) : = max { | B (S) | − β | S | s u c h t h a t S ⊆ V } . The case in which β = 1 is known as the differential of G, and hence ∂ β (G) can be considered as a generalization of the differential ∂ (G) of G. In this paper, upper and lower bounds for ∂ β (G) are given in terms of its order | G | , minimum degree δ (G) , maximum degree Δ (G) , among other invariants of G. Likewise, the β -differential for graphs with heavy vertices is studied, extending the set of applications that this concept can have. [ABSTRACT FROM AUTHOR]

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]