Let $P$ be a set of $n$ points in the plane, and let $C$ be a compact, convex subset of the plane that contains the origin in its interior. We study the problem of finding two points $p$ and $q$ of $P$ with the property that any homothet of $C$ that contains $p$ and $q$ must contain at least a constant fraction of the points of $P$. We give proofs for several cases. When $C$ is strictly convex we achieve a fraction of $1/5$, and when $C$ is a general convex region we achieve a fraction of $1/6e$. We show how the ideas of this second bound can be used to prove a lower bound for a version of the problem in $\mathbb{R}^d$ when $C$ is a $d$-ball. We also consider a few special cases. When $C$ is a triangle we get a lower bound of $1/12$, and when we allow that triangle to be rotated by $π$ radians in addition to being scaled and translated we get a lower bound of $1/768$. In proving this last result, we improve the state of the art on the version of this problem in three dimensions, where $C$ is a parallelepiped. We also consider the algorithmic aspects of finding such a pair of points.
Prosenjit Bose, Vida Dujmović, Hussein Houdrouge, Pat Morin, Saeed Odak
原文摘要
A dominating set of a graph $G$ is \emph{connected} if it induces a connected graph in $G$. For planar triangulations, it has been known since 1990 that every $n$-vertex triangulation with $n\ge4$ admits a connected dominating set of size at most $n/2 - 1$, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every triangulation admits a spanning tree with at least $11n/21$ leaves. This result has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings. The connected dominating set results extend readily to show that an $n$-vertex triangulation of a surface of Euler genus $g$ has a connected dominating set of size at most $10n/21+O(\sqrt{gn})$.