Playing with parameters: structural parameterization in graphs

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” parameter). A natural subject for investigation is what happens when we parameterize such a problem by various other parameters, some of which may be the values of optimal solutions to different problems. Such research is known as parameterized ecology. In this paper, 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.