The Complexity of Contracting Bipartite Graphs into Small Cycles

For a positive integer l >= 3, the C-l-CONTRACTIBILITY problem takes as input an undirected simple graph G and determines whether G can be transformed into a graph isomorphic to C-l (the induced cycle on l vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that C-4-CONTRACTIBILITY is NP-complete in general graphs. It is easy to verify that that C-3-CONTRACTIBILITY is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that C-l-CONTRACTIBILITY is NP-complete on bipartite graphs for l = 6 and posed as open problems the status of C-l-CONTRACTIBILITY when l is 4 or 5. In this paper, we show that both C-5-CONTRACTIBILITY and C-4-CONTRACTIBILITY are NP-complete on bipartite graphs.