  Eternal Vertex Cover problem is a dynamic variant of the vertex cover
problem. We have a two player game in which guards are placed on some vertices
of a graph. In every move, one player (the attacker) attacks an edge. In
response to the attack, the second player (defender) moves the guards along the
edges of the graph in such a manner that at least one guard moves along the
attacked edge. If such a movement is not possible, then the attacker wins. If
the defender can defend the graph against an infinite sequence of attacks, then
the defender wins.
  The minimum number of guards with which the defender has a winning strategy
is called the Eternal Vertex Cover Number of the graph G. On general graphs,
the computational problem of determining the minimum eternal vertex cover
number is NP-hard and admits a 2-approximation algorithm and an exponential
kernel. The complexity of the problem on bipartite graphs is open, as is the
question of whether the problem admits a polynomial kernel.
  We settle both these questions by showing that Eternal Vertex Cover is
NP-hard and does not admit a polynomial compression even on bipartite graphs of
diameter six. We also show that the problem admits a polynomial time algorithm
on the class of cobipartite graphs.
