🤖 AI Summary
This paper investigates the $C_4$- and $C_5$-Contractibility problems on bipartite graphs—i.e., whether a given bipartite graph can be transformed into a cycle of length 4 or 5 via edge contractions—resolving an open problem posed by Dabrowski and Paulusma. While $C_3$-Contractibility is known to be polynomial-time solvable and $C_6$-Contractibility NP-complete on bipartite graphs, the complexity for $ell = 4,5$ had remained unresolved. We prove that both $C_4$- and $C_5$-Contractibility are NP-complete on bipartite graphs. Our proof employs carefully constructed polynomial-time reductions: from 3-SAT for $C_4$-Contractibility and from One-in-Three 3-SAT for $C_5$-Contractibility. These reductions leverage structural properties of bipartite graphs and precise constraints imposed by edge contraction to ensure correctness and feasibility. This work completes the complexity landscape of $C_ell$-Contractibility on bipartite graphs for $ell = 3$–$6$, thereby filling a critical gap in the understanding of cycle contractibility.
📝 Abstract
For a positive integer $ell geq 3$, the $C_ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_ell$ (the induced cycle on $ell$ 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 $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{ell}$-Contractibility is NP-complete on bipartite graphs for $ell = 6$ and posed as open problems the status of the problem when $ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs.
