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On the largest dynamic monopolies of graphs with a given average threshold

Let $G$ be a graph and $τ$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a $τ$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and for any $i\in \{0, \ldots, k-1\}$, each vertex $v$ in $D_{i+1}$ has at least $τ(v)$ neighbors in $D_0\cup \ldots \cup D_i$. Denote the size of smallest $τ$-dynamic monopoly by $dyn_τ(G)$ and the average of thresholds in $τ$ by $\overlineτ$. We show that the values of $dyn_τ(G)$ over all assignments $τ$ with the same average threshold is a continuous set of integers. For any positive number $t$, denote the maximum $dyn_τ(G)$ taken over all threshold assignments $τ$ with $\overlineτ\leq t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worst-case value of a dynamic monopoly when the average threshold is a given number $t$. We investigate under what conditions on $t$, there exists an upper bound for $Ldyn_{t}(G)$ of the form $c|G|$, where $c<1$. Next, we show that $Ldyn_t(G)$ is coNP-hard for planar graphs but has polynomial-time solution for forests.