arXiv:2312.03399 · v3
Connected Dominating Sets in Triangulations
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 $10n/21+O(\sqrt{gn})$.
作者备注：Updated version with several corrections and improvements. Most notably, the use of potential functions rather than linear programs to bound the sizes of dominating sets