What is the treewidth of the icosahedral plane triangulation? I deleted my previously posted question because it seemed not well received, likely because I didn’t flesh it out sufficiently.  Let me try again.
We can draw the plane triangulation of the icosahedron as shown in the following diagram.
We know that the treewidth of a subgraph is a lower bound on the treewidth of the entire graph.  So, let’s focus first on the subgraph of order $9$ induced by {$A, B, C, D, E, F, X, Y, Z$}. Let’s take {$X,Y,Z,F$} as the “anchor” for the following “smooth” tree decomposition (actually also a “path” decomposition) of this subgraph:
$X_1 = ${$X, Y, Z, F, A, B$}
$X_2 = ${$X, Y, Z, F, B, C$}
$X_3 = ${$X, Y, Z, F, C, D$}
$X_4 = ${$X, Y, Z, F, D, E$}
The width of this decomposition is $5$, one less than the number of vertices in each $X_i$. This establishes an upper bound on the treewidth of the subgraph.  So, the questions I have are:
1:  What is the tree decomposition of the subgraph that has the smallest width (namely, a decomposition that establishes the treewidth of the subgraph)?
2:  What is the tree decomposition of the entire icosahedral triangulation that has the smallest width?
With respect to question 2, we get a trivial upper bound of $8$ on the treewidth of the icosahedron because we can simply add vertices {$1, 2, 3$} to each of the $X_i$ to get a smooth decomposition of the icosahedron.  But surely, we can do much better than that.

 A: The following tree decomposition for the subgraph that is the subject of question 1 has width $4$, thus establishing that as an upper limit on the treewidth for the subgraph. The remaining issue for question 1 is whether there is a tree decomposition with width $3$ (below which one can most certainly not go). And question 2 is still open.
$X_1 =$ {$F, Z, X, A, B$}
$X_2 =$ {$F, Z, X, C, B$}
$X_3 =$ {$F, Z, X, C, Y$}
$X_4 =$ {$F, Z, D, C, Y$}
$X_5 =$ {$F, Z, D, E, Y$}
So, after working on this a while longer, I cannot find an improvement to my answer to question 1. And, with respect to question 2, the best I can find is a "smooth" tree decomposition (that is also a "path" decomposition") of width 6, suggesting that the treewidth of the icosahedron is 6.  Following is that decomposition:
$X_1 =$ {$1, 2, F, Z, X, A, B$}
$X_2 =$ {$1, 2, F, Z, X, C, B$}
$X_3 =$ {$1, 2, F, Z, X, C, Y$}
$X_4 =$ {$1, 2, F, Z, D, C, Y$}
$X_5 =$ {$1, 2, F, Z, D, 3, Y$}
$X_6 =$ {$1, E, F, Z, D, 3, Y$}
Notice that for both the subgraph and the entire graph, the smooth decompositions form paths where each successive node is obtained from the previous by a substitution of a single vertex for another.  This is a general property.
I believe these to be the correct answers, but because this stuff is tricky, I am not absolutely sure.  I would like to hear from anyone who determines the treewidths to be smaller than I have found.
