  This work studies a dynamic, adversarial resource allocation problem in
environments modeled as graphs. A blue team of defender robots are deployed in
the environment to protect the nodes from a red team of attacker robots. We
formulate the engagement as a discrete-time dynamic game, where the robots can
move at most one hop in each time step. The game terminates with the attacker's
win if any location has more attacker robots than defender robots at any time.
The goal is to identify dynamic resource allocation strategies, as well as the
conditions that determines the winner: graph structure, available resources,
and initial conditions. We analyze the problem using reachable sets and show
how the outdegree of the underlying graph directly influences the difficulty of
the defending task. Furthermore, we provide algorithms that identify
sufficiency of attacker's victory.
