When considering a graph problem from a parameterized point of view, the parameter chosen is often the size of an optimal solution of this problem (the "standard"). A natural subject for investigation is what happens when we parameterize such a problem by the size of an optimal solution of a different problem. We provide a framework for doing such analysis. In particular, we investigate seven natural vertex problems, along with their respective parameters: α (the size of a maximum independent set), τ (the size of a minimum vertex cover), ω (the size of a maximum clique), χ (the chromatic number), γ (the size of a minimum dominating set), i (the size of a minimum independent dominating set) and ν (the size of a minimum feedback vertex set). We study the parameterized complexity of each of these problems with respect to the standard parameter of the others.
