Flips in Odd Matchings

Abstract

Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in $GM_\mathcal{P}$ are connected if the two corresponding matchings have $m-1$ edges in common. In this work we show that $GM_\mathcal{P}$ is connected and give an upper bound of $O(n^2)$ on its diameter. Moreover, we present a tight bound of $Θ(n)$ for the diameter of the flip graph of points in convex position.

Figures (6)

• Figure 1: Flipping a matching edge: the previously unmatched point $p$ is matched to $q$.
• Figure 2: A plane alternating path in the visibility graph gives rise to a sequence of flips.
• Figure 3: Constructing $G_{k+1}$ (right) from $G_k$ (left). The paths $G_k$ and $G_{k+1}$ are depicted with lines, while unused edges of $G$ are dashed. The matching edges are red, the cycle edges are black.
• Figure 4: It takes $\Omega(n)$ flips to transform the given matching to any matching where the unmatched point is on the boundary of the convex hull.
• Figure 5: To flip from matching $M_1$ (left, black) to matching $M_2$ (right, black), $m-2$ horizontal edges (all except the two bottom most) need to be flipped twice. The red edges give a shortest flip sequence, which takes $2m$ edge flips.

Theorems & Definitions (10)

• Theorem 1
• Lemma 1
• Lemma 1
• proof : Proof of \ref{['thm:flip_graph_connected']}
• Lemma 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof