On the Hardness of Eliminating Small Induced Subgraphs by Contracting Edges

Graph modification problems such as vertex deletion, edge deletion or edge contractions are a fundamental class of optimization problems. Recently, the parameterized complexity of the contractibility problem has been pursued for various specific classes of graphs. Usually, several graph modification questions of the deletion variety can be seen to be FPT if the graph class we want to delete into can be characterized by a finite number of forbidden subgraphs. For example, to check if there exists k vertices/edges whose removal makes the graph C4-free, we could simply branch over all cycles of length four in the given graph, leading to a search tree with O(4) leaves. Somewhat surprisingly, we show that the corresponding question in the context of contractibility is in fact W[2]-hard. An immediate consequence of our reductions is that it is W[2]-hard to determine if at most k edges can be contracted to modify the given graph into a chordal graph. More precisely, we obtain following results: • Cl-free Contraction is W[2]-hard if l ⩾ 4 and FPT if l ⩽ 3. • Pl-free Contraction is W[2]-hard if l ⩾ 5 and FPT if l ⩽ 4, where Pl denotes a path on l vertices. We believe that this opens up an interesting line of work in understanding the complexity of contractibility from the perspective of the graph classes that we are modifying into.