This paper deals with the detection and localization of damage in gas-filled pipelines by means of an oriented-signals graph. The purpose is to search for transient patterns in a faulty main conduit, when an external acoustic source excites the viscous fluid. The study assumes a linearized model of distributed parameters for the healthy sections of the pipe and two-port lumped models for leaks and blockages. Based on these models, the exact transcendental transfer function of an acoustic source at the upstream end P(0,s) to the pressure P(z0,s) at point z0 is derived and transformed through graph tools in an oriented-signals graph for one and two faults. Thus, one establishes the transient response patterns associated with the faults from the graph paths without numerical parameters. This shown that graph theory is another suitable tool to visualize pressure wave propagation in a damage conduit. To validate the responses for one and two leaks, one compares the features with the analytical solution and the digital simulated pressure.