Matching in Gabriel Graphs

Abstract

Given a set $P$ of $n$ points in the plane, the order-$k$ Gabriel graph on $P$, denoted by $k$-$GG$, has an edge between two points $p$ and $q$ if and only if the closed disk with diameter $pq$ contains at most $k$ points of $P$, excluding $p$ and $q$. We study matching problems in $k$-$GG$ graphs. We show that a Euclidean bottleneck perfect matching of $P$ is contained in $10$-$GG$, but $8$-$GG$ may not have any Euclidean bottleneck perfect matching. In addition we show that $0$-$GG$ has a matching of size at least $\frac{n-1}{4}$ and this bound is tight. We also prove that $1$-$GG$ has a matching of size at least $\frac{2(n-1)}{5}$ and $2$-$GG$ has a perfect matching. Finally we consider the problem of blocking the edges of $k$-$GG$.