TREEWIDTH AND GONALITY OF GLUED GRID GRAPHS IVAN AIDUN, FRANCES DEAN, RALPH MORRISON, TERESA YU, AND JULIE YUAN Abstract. We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs. 1. Introduction The treewidth of a graph is a measure of the graph's similarity to a tree, first introduced by Robertson and Seymour in [14]. Treewidth is a natural and powerful measure of a graph's complexity: there are polynomial time algorithms for many difficult problems on graphs of bounded treewidth [4]. A well-known result is that an n × n grid graph has treewidth n [5], implying that the treewidth of planar graphs is unbounded; this is in contrast to similar graph parameters such as the Hadwiger number, which is bounded for planar graphs [19]. Since treewidth is minor-monotone [8], the treewidth of a graph is at least the size of its largest grid minor. For these reasons, studying grid graphs is important in the study of treewidth. Figure 1. The 4 × 3 grid, stacked prism, and toroidal grid, respectively In this paper, we are interested in certain natural generalizations of the grid graphs, which we collectively refer to as the glued grids: the stacked prism graphs Ym;n = Cm Pn and the toroidal grid graphs Tm;n = Cm Cn. Here Pn is the path on n vertices, Cn is the cycle on n vertices, and G H denotes the Cartesian product of G and H. As illustrated in Figure 1, the glued grids resemble grids with additional edges: both the stacked prisms and the toroidal grids have edges wrapping from the top row to the bottom row, and the toroidal grids also have edges wrapping from the leftmost column to the rightmost. In this sense, we can see them as grids glued along their boundaries, analogous to topological quotients on the boundary of the unit square. It was shown in [10] that the treewidth of the n × n toroidal grid is either 2n − 2 or 2n − 1, and that both of these values are achieved for certain n. They also give the following general bounds: minfm; ng ≤ tw(Ym;n) ≤ minfm; 2ng arXiv:1808.09475v2 [math.CO] 2 Nov 2019 and 2 minfm; ng − 2 ≤ tw(Tm;n) ≤ 2 · minfm; ng: We use strict brambles to explicitly compute the treewidth of glued grids in all but three exceptional cases, namely Y2n;n, Tn+1;n, and Tn;n, the third of which was already studied in [10]. Our two main treewidth results are the following, with the main result of [10] included as the third case of Theorem 1.2. Theorem 1.1. The treewidth of the stacked prism Ym;n is ( minfm; 2ng if m 6= 2n tw(Ym;n) = minfm; 2ng − 1 or minfm; 2ng if m = 2n: 1 2 TREEWIDTH AND GONALITY OF GLUED GRID GRAPHS If m = 2n, both possible treewidths occur for certain values of n. Theorem 1.2. The treewidth of the toroidal grid Tm;n is 8 2 minfm; ng if jm − nj ≥ 2 <> tw(Tm;n) = 2 minfm; ng − 1 or 2 minfm; ng if jm − nj = 1 :>2n − 2 or 2n − 1 if m = n: If jm − nj = 1 or m = n, both possible treewidths occur for certain values of m and n. Our original motivation for this project was to understand divisorial gonality, another invariant of these glued graphs. In Section 5, we discuss connections between gonality and treewidth, and we compute the gonality of all glued grids except for Y2n;n, Tn+1;n, and Tn;n in Theorems 5.2 and 5.3. Our paper is organized as follows. In Section 2, we review background and definitions and establish an explicit link between treewidth and strict brambles. In Sections 3 and 4, we prove Theorems 1.1 and 1.2, respectively. Section 5 consists of our results on divisorial gonality. 2. Treewidth, brambles, and strict brambles Throughout this paper we let G = (V; E) be a connected, simple graph with vertex set V and edge set E. When multiple graphs are under discussion, we use V (G) and E(G) to denote the vertices and edges of G, respectively. A subset A ⊂ V (G) is said to be connected if the subgraph it induces is a connected graph. Given two graphs G1 = (V1;E1) and G2 = (V2;E2), the Cartesian product G1 G2 is the graph with vertex set V (G1 × G2) = V1 × V2 and an edge between (v1; v2) and (u1; u2) if • v2 = u2 and v1u1 2 E1, or • v1 = u1 and v2u2 2 E2. With this definition, we see that the m × n grid graph Gm;n is the Cartesian product Pm Pn of two path graphs; the stacked prism graph Ym;n is the Cartesian product Cm Pn of a cycle graph and a path; and the toroidal grid graph Tm;n is the Cartesian product Cm Cn of two cycles. To ensure these graphs are simple, we assume m ≥ 3 for Ym;n and that m; n ≥ 3 for Tm;n. Given a graph G, let T be a tree, and let V = fVtg be a family of subsets Vt ⊂ V (G) indexed by the nodes t 2 T . The pair (T; V) constitutes a tree decomposition of G if it satisfies the following: [ (1) Vt = V (G); t2T (2) if uv 2 E(G) then there is a node t 2 T such that u; v 2 Vt; and 0 0 (3) if v 2 Vt1 ;Vt2 , then for every t in the (unique) path from t1 to t2 in T , we have v 2 Vt . The sets Vt are commonly referred to as bags. The width of a tree decomposition is one less than the size of the largest bag: w(T; V) = maxt2T (jVtj − 1). The treewidth of G, denoted tw(G), is the minimum width of a tree decomposition of G. One useful property of a tree decomposition is its connection to separating sets in the starting graph. Given subsets S, U1; and U2 of V (G), we say S separates U1 from U2 if every path from an element of U1 to an element of U2 includes an element of S. Note that if S separates U1 and U2, then U1 \ U2 ⊆ S. Lemma 2.1 (Lemma 12.3.1 in [6]). Let (T; V) be a tree decomposition of a graph G. Let t1t2 be any edge of T , and let T and T be the two connected components of T − t t , where t 2 T . If U = S V , then V \ V 1 2 1 2 i i i t2Ti t t1 t2 separates U1 from U2 in G. Our main tools for computing treewidth are structures called brambles. We say two subsets A; B ⊂ V (G) touch if either they share a vertex or there is an edge between them. A bramble is a family of connected, mutually touching vertex sets. If for all B1;B2 2 B we have B1 \ B2 6= ;, then the bramble B is called strict. A set S ⊂ V (G) is said to cover the bramble B if S \ B 6= ; for all B 2 B. A set which covers B is called a hitting set for B. The order of a bramble B, written kBk, is the minimum size of a hitting set for B. Example 2.2 (From Section 4.5 in [17]). Consider the m × n grid Gm;n. Let B = fBijg, where Bij is the union th th th of the i column with the j row. This is a strict bramble: Bij and Bk` intersect at the intersection of the i column with the `th row. It turns out that kBk = minfm; ng. To see this, assume without loss of generality that m ≤ n. If S consists of the m vertices in a column, then S is a hitting set, as each Bij intersects every column in TREEWIDTH AND GONALITY OF GLUED GRID GRAPHS 3 at least one point. So, kBk ≤ m. However, any set T with jT j ≤ m − 1 is not a hitting set, as T must then miss some row and some column, and so there exists some Bij such that Bij \ T = ;. It follows that kBk ≥ m, and we conclude that kBk = m = minfm; ng. The bramble number (resp. strict bramble number) of a graph is the maximum order of a bramble (resp. strict bramble) on G. We denote the bramble number bn(G) and the strict bramble number sbn(G). We can immediately see that sbn(G) ≤ bn(G), since every strict bramble is a bramble. Example 2.3. For an m×n grid graph, we have bn(Gm;n) = tw(Gm;n)+1 = minfm; ng+1 by [5] and Proposition 2.5 below. On the other hand, sbn(Gm;n) ≥ minfm; ng by Example 2.2. Since sbn(Gm;n) ≤ bn(Gm;n), the bramble number and the strict bramble number differ by at most 1. (It will follow from Lemma 2.6 that they cannot be equal, so they differ by exactly 1.) It was shown in [11, x2.2] that bn(G) ≤ 2 sbn(G), so the bramble number and the strict bramble number never differ by more than a factor of 2. This bound is sharp, due to the following example.