Concetti Chiave
Every n-vertex triangulation has a connected dominating set of size at most 10n/21.
Sintesi
The paper presents algorithms for efficiently constructing small connected dominating sets in triangulations. The key results are:
- Every n-vertex triangulation has a connected dominating set of size at most 10n/21. This improves upon the previous best bound of n/2.
- Every n-vertex triangulation has a spanning tree with at least 11n/21 leaves. This provides progress on the maxleaf spanning-tree problem for triangulations.
- The results extend to n-vertex triangulations of genus-g surfaces, where the connected dominating set size is at most 10n/21 + O(√gn).
The algorithms work by incrementally growing a connected dominating set in small batches. The main technical challenges are:
- Identifying and handling "critical" subgraphs where vertices have inner-degree at most 2.
- Efficiently dominating degree-1 and degree-0 vertices in the critical subgraphs.
- Carefully constructing a sequence of sets that satisfy the properties required for the final connected dominating set.
The paper also discusses connections to the problem of finding one-bend free sets in planar graphs, showing that the leaves of the spanning tree provide a one-bend free set of size at least 11n/21.
Statistiche
Every n-vertex triangulation has a dominating set of size at most n/3.
Every n-vertex triangulation has a connected dominating set of size at most 2n/3.
Every n-vertex triangulation has a spanning tree with at least (n+2)/2 leaves.
Citazioni
"For every n ≥3, every n-vertex triangulation G has a connected dominating set X of size at most 10n/21 = 0.476190n."
"For every n ≥3, every n-vertex triangulation G has a spanning tree T with at least 11n/21 = 0.523809n leaves."