Consider an information network with harmful procedures called attackers (e.g., viruses); each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is the system protector scanning and cleaning from attackers some part of the network (e.g., an edge or a path), which it chooses independently using another probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the system protector; towards a conflicting objective, the system protector aims at maximizing the expected number of cleaned attackers. We model this network scenario as a non-cooperative strategic game on graphs. We focus on the special case where the protector chooses a single edge. We are interested in the associated Nash equilibria, where no network entity can unilaterally improve its local objective. We obtain the following results: âï申ï申 No instance of the game possesses a pure Nash equilibrium. âï申ï申Every mixed Nash equilibrium enjoys a graph-theoretic structure, which enables a (typically exponential) algorithm to compute it. âï申ï申 We coin a natural subclass of mixed Nash equilibria, which we call matching Nash equilibria, for this game on graphs. Matching Nash equilibria are defined using structural parameters of graphs, such as independent sets and matchings. âï申ï申We derive a characterization of graphs possessing matching Nash equilibria. The characterization enables a linear time algorithm to compute a matching Nash equilibrium on any such graph with a given independent set and vertex cover. âï申ï申 Bipartite graphs are shown to satisfy the characterization. So, using a polynomial-time algorithm to compute a perfect matching in a bipartite graph, we obtain, as our main result, an efficient graph-theoretic algorithm to compute a matching Nash equilibrium on any instance of the game with a bipartite graph.