  In the deterministic binary majority process we are given a simple graph
where each node has one out of two initial opinions. In every round, every node
adopts the majority opinion among its neighbors. By using a potential argument
first discovered by Goles and Olivos (1980), it is known that this process
always converges in $O(|E|)$ rounds to a two-periodic state in which every node
either keeps its opinion or changes it in every round.
  It has been shown by Frischknecht, Keller, and Wattenhofer (2013) that the
$O(|E|)$ bound on the convergence time of the deterministic binary majority
process is indeed tight even for dense graphs. However, in many graphs such as
the complete graph, from any initial opinion assignment, the process converges
in just a constant number of rounds.
  By carefully exploiting the structure of the potential function by Goles and
Olivos (1980), we derive a new upper bound on the convergence time of the
deterministic binary majority process that accounts for such exceptional cases.
We show that it is possible to identify certain modules of a graph $G$ in order
to obtain a new graph $G^\Delta$ with the property that the worst-case
convergence time of $G^\Delta$ is an upper bound on that of $G$. Moreover, even
though our upper bound can be computed in linear time, we show that, given an
integer $k$, it is NP-hard to decide whether there exists an initial opinion
assignment for which it takes more than $k$ rounds to converge to the
two-periodic state.
