How to find all connections in the Pantelides algorithm for delay differential-algebraic equations

The Pantelides algorithm for delay differential-algebraic equations (DDAEs) is a method to structurally analyse such systems with the goal to detect which equations have to be differentiated or shifted to construct a solution. In this process, one has to detect implicit connections between equations in the shifting graph, making it necessary to check all possible connections. The problem of finding these efficiently remained unsolved so far. It is explored in further detail and a reformulation is introduced. Additionally, an algorithmic approach for its solution is presented.