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Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph G are contaminated initially, before the process starts. By the q -forcing rule, a contaminated vertex having at most q uncontaminated neighbors enforces all the neighbors to become contaminated, while by the p -percolation rule, an uncontaminated vertex becomes contaminated if at least p of its neighbors are contaminated. If given a set S of initially contaminated vertices all vertices eventually become contaminated when continuously applying the q -forcing rule (respectively the p -percolation rule), S is called a q -forcing set (respectively, a p -percolating set) in G. In this paper, we consider sets S that are at the same time q -forcing sets and p -percolating sets, and call them (p , q) -spreading sets. Given positive integers p and q , the minimum cardinality of a (p , q) -spreading set in G is a (p , q) -spreading number, σ (p , q) (G) , of G. While q -forcing sets have been studied in a dozen of papers, the decision version of the corresponding graph invariant has not been considered earlier, and we fill the gap by proving its NP-completeness. This, in turn, enables us to prove the NP-completeness of the decision version of the (p , q) -spreading number in graphs for an arbitrary choice of p and q. Again, for every p ∈ N and q ∈ N ∪ { ∞ } , we find a linear-time algorithm for determining the (p , q) -spreading number of a tree, where in the case p ≥ 2 we apply Riedl's algorithm from [Largest and smallest minimal percolating sets in trees, Electron. J. Combin. 19 (2012) Paper 64] on p -percolation in trees. In addition, we present a lower and an upper bound on the (p , q) -spreading number of a tree and characterize extremal families of trees. In the case of square grids, we combine some results of Bollobás from [The Art of Mathematics: Coffee Time in Memphis. Cambridge Univ. Press, New York, 2006], and the AIM Minimum Rank-Special Graphs Work Group from [Zero forcing sets and the minimum rank of graphs, Linear algebra Appl. 428 2008 1628–1648], and new results on (2 , 1) -spreading and (4 , q) -spreading to obtain σ (p , q) (P m □ P n) for all (p , q) ∈ (N ∖ { 3 }) × (N ∪ { ∞ }) and all m , n ∈ N. [ABSTRACT FROM AUTHOR]