Triangulations Admit Dominating Sets of Size $2n/7$

We show that every planar triangulation on $n>10$ vertices has a dominating set of size $n/7=n/3.5$. This approaches the $n/4$ bound conjectured by Matheson and Tarjan [MT'96], and improves significantly on the previous best bound of $17n/53\approx n/3.117$ by \v{S}pacapan [\v{S}'20]. From our proof it follows that every 3-connected $n$-vertex near-triangulation (except for 3 sporadic examples) has a dominating set of size $n/3.5$. On the other hand, for 3-connected near-triangulations, we show a lower bound of $3(n-1)/11\approx n/3.666$, demonstrating that the conjecture by Matheson and Tarjan [MT'96] cannot be strengthened to 3-connected near-triangulations. Our proof uses a penalty function that, aside from the number of vertices, penalises vertices of degree 2 and specific constellations of neighbours of degree 3 along the boundary of the outer face. To facilitate induction, we not only consider near-triangulations, but a wider class of graphs (skeletal triangulations), allowing us to delete vertices more freely. Our main technical contribution is a set of attachments, that are small graphs we inductively attach to our graph, in order both to remember whether existing vertices are already dominated, and that serve as a tool in a divide and conquer approach. Along with a well-chosen potential function, we thus both remove and add vertices during the induction proof. We complement our proof with a constructive algorithm that returns a dominating set of size $\le 2n/7$. Our algorithm has a quadratic running time.