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Free keywords:
Computer Science, Computational Complexity, cs.CC
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.
