  In this article, we start with a two-player game that models communication
under adverse circumstances in everyday life and study it from the perspective
of a modal logic of graphs, where links can be deleted locally according to
definitions available to the adversarial player. We first introduce a new
language, semantics, and some typical validities. We then formulate a new type
of first-order translation for this modal logic and prove its correctness.
Then, a novel notion of bisimulation is proposed which leads to a
characterization theorem for the logic as a fragment of first-order logic, and
a further investigation is made of its expressive power against hybrid modal
languages. Next, we discuss how to axiomatize this logic of link deletion,
using dynamic-epistemic logics as a contrast. Finally, we show that our new
modal logic lacks both the tree model property and the finite model property,
and that its satisfiability problem is undecidable.
