In the domination game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set A * dominates the whole graph. The Dominator plays to minimise the size of the set A * while the Staller plays to maximise it. A graph is D-trivial if when the Dominator plays first and both players play optimally, the set A * is a minimum dominating set of the graph. A graph is S-trivial if the same is true when the Staller plays first. We consider the problem of characterising D-trivial and S-trivial graphs. We give complete characterisations of D-trivial forests and of S-trivial forests. We also show that 2-connected D-trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.

Let G be a graph and τ : V (G) → N ∪ {0} be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a dynamic monopoly corresponding to (G, τ ) if the vertices of G can be partitioned into subsets D 0 , D 1 , . . . , D k such that D 0 = D and for any i ∈ {0, . . . , k − 1}, each vertex v in D i+1 has at least τ (v) neighbors in D 0 ∪ . . . ∪ D i . Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of any dynamic monopoly in terms of the average threshold and the order of graph. Also we obtain an upper bound in terms of the minimum vertex cover of graphs. Then we derive the upper bound |G|/2 for the smallest size of any dynamic monopoly when the graph G contains at least one odd vertex, where the threshold of any vertex v is set as ⌈(deg(v) + 1)/2⌉ (i.e. strict majority threshold). This bound improves the best known bound for strict majority threshold. We show that the latter bound can be achieved by a polynomial time algorithm. We also show that α ′ (G) + 1 is an upper bound for the size of strict majority dynamic monopoly, where α ′ (G) stands for the matching number of G. Finally, we obtain a basic upper bound for the smallest size of any dynamic monopoly, in terms of the average threshold and vertex degrees. Using this bound we derive some other upper bounds.Mathematics Subject Classification: 91D30, 05C85, o5C69.

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Irreversible dynamic monopolies were already defined and widely studied in the literature for undirected graphs. They are arising from formulation of the irreversible spread of influence such as disease, opinion, adaptation of a new product, etc., in social networks, where the influence between any two individuals is assumed to be bilateral or reciprocal. But in many phenomena, the influence in the underlying network is unilateral or one-sided. In order to study the latter models we need to introduce and study the concept of dynamic monopolies in directed graphs. Let G be a directed graph such that the in-degree of any vertex G is at least one. Let also τ : V (G) → N be an assignment of thresholds to the vertices of G. A subset M of vertices of G is called a dynamic monopoly for (G, τ ) if the vertex set of G can be partitioned into D 0 ∪ . . . ∪ D t such that D 0 = M and for any i ≥ 1 and any v ∈ D i , the number of edges from D 0 ∪ . . . ∪ D i−1 to v is at least τ (v). One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment in which for any vertex v, τ (v) = ⌈(deg in (v) + 1)/2⌉, where deg in (v) stands for the in-degree of v. By a strict majority dynamic monopoly of a graph G we mean any dynamic monopoly of G with strict majority threshold assignment for the vertices of G. In this paper we first discuss some basic upper and lower bounds for the size of dynamic monopolies with general threshold assignments and then obtain some hardness complexity results concerning the smallest size of dynamic monopolies in directed graphs. Next we show that any directed graph on n vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with n/2 vertices. * Corresponding author: mzaker@iasbs.ac.irWe show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the best known result. The final note of the paper deals with the possibility of the improvement of the latter n/2 bound.Mathematics Subject Classification: 05C20, 05C82, 91D30