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The aim of this paper is to analyze the relationship between marginality and the Myerson value, the within groups Myerson value (WG-Myerson value) and the between groups Myerson value (BG-Myerson value). We enlarge the idea of the classical marginal contribution of a player to a coalition in a coopera- tive game. Besides this type of contribution, in games with cooperation restricted by a graph, a player can contribute to a coalition in other ways. For example, lending his links to the coalition but without joining it. We will call it the marginal contribution of the player's links (L-marginal contribution). Also he can contribute to a coalition by joining it with his communication possibilities. This is the marginal contribu- tion of the player with his links (PL-marginal contribution). According to this, we define the strong mono- tonicity of the allocation rules with respect to the L-marginal contributions (and the L-marginality); and similarly, the strong monotonicity with respect to the PL-marginal contributions (and the PL-marginality). We prove that the Myerson value, the WG-Myerson value and the BG-Myerson value can be character- ized using as requirement PL-marginality, marginality and L-marginality, respectively (as well as other properties).