Speakers
Description
Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shunichi Maezawa, Yuta Nozaki, Yoshio Okamoto and Kenta Ozeki
Abstract: In this paper, we consider a transformation of $k$ disjoint paths in a graph. For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$ connecting the same set of terminal pairs, we aim to determine whether $\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one path with another path so that the intermediates are also $k$ disjoint paths. The problem is called \textsc{Disjoint Paths Reconfiguration}. We first show that \textsc{Disjoint Paths Reconfiguration} is $PSPACE$-complete even when $k=2$. On the other hand, we prove that, when the graph is embedded on a plane and all paths in $\mathcal{P}$ and $\mathcal{Q}$ connect the boundaries of two faces, \textsc{Disjoint Paths Reconfiguration} can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint $s$-$t$ paths as a variant. We show that the disjoint $s$-$t$ paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is $PSPACE$-complete in general.