Abstract
Let $G$ be a complete convex geometric graph whose vertex set $P$ forms a convex polygon $C$, and let $\mathcal{F}$ be a family of subgraphs of $G$. A blocker for $\mathcal{F}$ is a set of diagonals of $C$, of smallest possible size, that contains a common edge with every element of $\mathcal{F}$. Previous works determined the blockers for various families $\mathcal{F}$ of non-crossing subgraphs, including the families of all perfect matchings, all spanning trees, all Hamiltonian paths, etc.
 In this paper we present a complete characterization of the family $\mathcal{B}$ of blockers for the family $\mathcal{T}$ of triangulations of $C$. In particular, we show that $|\mathcal{B}|=F_{2n-8}$, where $F_k$ is the $k$'th element in the Fibonacci sequence and $n=|P|$.
 We use our characterization to obtain a tight result on a geometric Maker-Breaker game in which the board is the set of diagonals of a convex $n$-gon $C$ and Maker seeks to occupy a triangulation of $C$. We show that in the $(1:1)$ triangulation game, Maker can ensure a win within $n-3$ moves, and that in the $(1:2)$ triangulation game, Breaker can ensure a win within $n-3$ moves. In particular, the threshold bias for the game is $2$.
Highlights
Determining the size of the blockers for F is a natural Turan-type question, as it is equivalent to determining the maximal size of a convex geometric graph that is free of F
A blocking set B for triangulations in C is a subgraph of G which contains a common edge with each element of T
In Theorem 1 we present a full characterization of the blockers, proving that any blocker is, in some sense, a hybrid of these two canonical blocking sets
Summary
We note that unlike in the above results (blockers for perfect matchins, etc.), when we consider blockers for triangulations, we treat the basic geometric graph G as the set of all diagonals of C, and identify any triangulation with the set of diagonals it the electronic journal of combinatorics 27(4) (2020), #P4.12 contains These notations are clearly crucial, since trivially, any boundary edge of C is a blocker for T. The edge set consisting of δ and all diagonals crossing δ, is an inclusion-minimal blocking set that may contain up to 1 +.
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