Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other?
We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $α$-orientations of a graph $G$, in which every vertex $v$ has a specified outdegree $α(v)$, and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $α$-orientations of a planar graph $G$ is at most two is \NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope.
We also consider the dual question of the flip distance betwe en graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang, Qian, and Zhang.
Computing distances in braid-move graphs, higher Bruhat orders, and oriented-matroid mutation graphs is NP-hard
Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；Abelium d.o.o.（阿贝利乌姆有限公司）
强有向图$D$的Wiener指标定义为所有有序顶点对之间的距离之和。如果在$D$中不存在从$a$到$b$的路径，则通过将从顶点$a$到顶点$b$的距离定义为$0$，该定义已被扩展到不一定是强的有向图。Knor，uSkrekovski和Tepeh [关于定向图的Wiener指标的一些注记.应用程序\数学。计算。\ { \bf 273}]考虑了具有最大Wiener指数的图的方向。证明了对于给定的树$T$，Wiener指数最大的$T$的方向$D$总是包含一个顶点$v$，使得对于每个顶点$u$，在$D$中要么有一条$(u,v)$ -路，要么有一条$(v,u)$ -路.在本文中，我们反驳了猜想。我们还证明了寻找给定图的最大Wiener指数的方向的问题是NP-完全的，从而回答了Knor，uSkrekovski和Tepeh [Orientations of graphs with maximum Wiener index.离散应用程序\数学。211]。我们简要地讨论了相应的问题，找到一个方向的最小维纳指数给定的图，并证明了特殊情况下，如果一个给定的图$m$边缘有一个方向的维纳指数$m$可以解决在$n$的时间二次。
