Abstract: In this paper, a concept is introduced in order to model a class of problems in social networks. A social network consisting of positive and negative influential individuals is represented as a partitioned graph G=(V,E) with vertex set partition V=V⁺\cup V^-. A subset D\subseteq V^- is called a positive influencing set if every vertex  of  G has  in D\cup V⁺ at least half of its neighbors. The positive influencing number of G is the minimum cardinality of a positive influencing set. In the present paper, we show some sharp lower bounds for this parameter, and prove that this problem is NP-complete for partitioned bipartite graphs and partitioned chordal graphs, respectively.

Key words: total domination, signed total domination, positive influencing number, social networks