Abstract. We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. Whether the combinatorial (i.e., graph-theoretic) distance between vertices of the associahedron can be computed in polynomial time is a tantalizing and important open problem, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that an approach for this open problem is not promising if it is applicable to the generalized problem on graph associahedra. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless [Formula: see text]. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids.

In this article, 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 Disjoint Paths Reconfiguration . We first show that Disjoint Paths Reconfiguration is \(\mathsf{PSPACE}\) -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, 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 \(\mathsf{PSPACE}\) -complete in general.

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 Disjoint Paths Reconfiguration. We first show that 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, 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.

We initiate the study of k -edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k -edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k -edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k -edge-connected orientation if and only if G is 2k -edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k -edge-connected orientations of an undirected graph G is connected if G is (2k+2) -edge connected. This has been known to be true only when k=1 .

We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge-connectivity, and the final orientation is k-edge-connected. This yields an “edge-flip based” new proof of Nash-Williams' theorem: an undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge-connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k + 2)-edge-connected. This has been known to be true only when k = 1.