On the treewidth of toroidal grids

Treewidth and gonality of glued grid graphs

“…Thus, we know that for any bramble B, tw(G) ≥ B − 1, so we can lower bound treewidth by constructing a bramble of large order. Indeed, this is the main technique in [9]. The next claim shows that we can omit the −1 when the bramble we construct is strict.…”

“…Since the strict bramble number is a lower bound on treewidth by Lemma 2.5, Propositions 3.1 and 3.2 imply that tw(Y m,n ) ≥ sbn(Y m,n ) ≥ min{m, 2n}. Using the upper bound from [9], we have the equality tw(Y m,n ) = min{m, 2n}. Now assume m = 2n.…”

“…Now assume m = 2n. Proposition 3.3 combined with the upper bound from [9] gives us that 2n − 1 ≤ tw(Y 2n,n ) ≤ 2n.…”

“…We then present a (non-strict) bramble of order 2 min{m, n} in the case when |m − n| = 1. We combine these with upper bounds on treewidth previous work from [9] to achieve our desired results from Theorem 1.2.…”

“…We use strict brambles to explicitly compute the treewidth of glued grids in all but three exceptional cases, namely Y 2n,n , T n+1,n , and T n,n , the third of which was already studied in [9]. Our two main treewidth results are the following, with the main result of [9] included as the third case of Theorem 1.2. If |m − n| = 1 or m = n, both possible treewidths occur for certain values of m and n.…”