Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds

Håvard Bakke Bjerkevik, Joseph Dorfer, Linda Kleist, Torsten Ueckerdt, Birgit Vogtenhuber

Abstract

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set $P$ of $n$ points in general position in the plane, the flip graph $F(P)$ has a vertex for each non-crossing spanning tree on $P$ and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam$(F(P))$ for sets $P$ of $n$ points in convex position. The current best bounds are $\frac{14}{9}n-O(1) \leq$ diam$(F(P))<\frac{15}{9}n-3$ [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most $\frac{14}{9}(n-1)$ to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph $F(P)$ for $n$ points in convex position to $\frac{11}{7}n-o(n)$.

Flip Distance of Non-Crossing Spanning Trees: NP-Hardness and Improved Bounds

Abstract

We consider the problem of reconfiguring non-crossing spanning trees on point sets. For a set of points in general position in the plane, the flip graph has a vertex for each non-crossing spanning tree on and an edge between any two spanning trees that can be transformed into each other by the exchange of a single edge. This flip graph has been intensively studied, lately with an emphasis on determining its diameter diam for sets of points in convex position. The current best bounds are diam [Bjerkevik, Kleist, Ueckerdt, and Vogtenhuber; SODA 2025]. The crucial tool for both the upper and lower bound are so-called *conflict graphs*, which the authors stated might be the key ingredient for determining the diameter (up to lower-order terms). In this paper, we pick up the concept of conflict graphs and show that this tool is even more versatile than previously hoped. As our first main result, we use conflict graphs to show that computing the flip distance between two non-crossing spanning trees is NP-hard, even for point sets in convex position. Interestingly, the result still holds for more constrained flip operations, concretely, compatible flips (where the removed and the added edge do not cross) and rotations (where the removed and the added edge share an endpoint). Extending the line of research from [BKUV SODA25], we present new insights on the diameter of the flip graph. Their lower bound is based on a constant-size pair of trees, one of which is *stacked*. We show that if one of the trees is stacked, then the lower bound is indeed optimal up to a constant term, that is, there exists a flip sequence of length at most to any other tree. Lastly, we improve the lower bound on the diameter of the flip graph for points in convex position to .
Paper Structure (5 sections, 4 theorems, 3 figures, 1 table)

This paper contains 5 sections, 4 theorems, 3 figures, 1 table.

Table of Contents

  1. Outline.
  2. Related work
  3. Connectedness and Diameter.
  4. Computational complexity of shortest flip sequences and the happy edge property.
  5. Fundamental concepts: Edge pairs, conflict graphs, and blowups

Key Result

Theorem 1

For any set $P$ of $n \geq 3$ points in convex position, the flip graph $\mathcal{F}(P)$ of non-crossing spanning trees on $P$ has diameter at least $(11/7-o(1))n$.

Figures (3)

  • Figure 1: Reconfiguration of non-crossing spanning trees on convex point sets by edge exchange. The appearing edge is highlighted in red and the disappearing edge in gray. Can you complete the sequence to obtain the tree on the right? How many further flips are needed?
  • Figure 2: Figure reproduced from bjerkevik2024flippingnoncrossingspanningtrees. \ref{['fig:cut-openA']} Two non-crossing trees $T,T'$ on a circularly labeled point set in convex position and \ref{['fig:cut-openB']} its linear representation with $T$ above and $T'$ below the spine.
  • Figure 3: \ref{['fig:IntroPairsA']} Illustration of the gap-edge and edge-edge bijection. Examples of an above \ref{['fig:IntroPairsB']}, a below \ref{['fig:IntroPairsC']}, and a crossing pair \ref{['fig:IntroPairsD']}. Edges belonging to near-near pairs in $A$, $B$, and $C$ are colored green, orange, and purple, respectively (also in several other figures).

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4