Treewidth is a lower bound on graph gonality

We prove that the (divisorial) gonality of a finite connected graph is lower bounded by its treewidth. We show that equality holds for grid graphs and complete multipartite graphs. We prove that the treewidth lower bound also holds for \emph{metric graphs} by constructing for any positive rank divisor on a metric graph $\Gamma$ a positive rank divisor of the same degree on a subdivision of the underlying graph. Finally, we show that the treewidth lower bound also holds for a related notion of gonality defined by Caporaso and for stable gonality as introduced by Cornelissen et al.


Introduction
In [2], Baker and Norine developed a theory of divisors on finite graphs analogous to divisor theory for Riemann surfaces. In particular, they stated and proved a graph theoretical analogue of the classical Riemann-Roch theorem. An important parameter of a Riemann surface is its gonality: the minimum degree of a holomorphic map to the Riemann sphere. Alternatively, the gonality can be defined as the minimum degree of a positive rank divisor. This second definition has a direct combinatorial analogue called the (divisorial) gonality, dgon(G), of a graph G. By Baker's Specialization Lemma, the gonality of a smooth algebraic curve is bounded from below by the gonality of (a subdivision of) the dual graph associated to the curve (see Corollary 3.2 in [1]). Hence, lower bounds for graph gonality, such as the treewidth bound proved here, can be used to obtain results for algebraic curves. See [1,9,15] for background on the interplay between divisors on graphs, curves and tropical curves.
As observed in [2], there is also a close connection between divisor theory on graphs and the chip-firing game of Björner-Lovász-Shor [5] (see also [17] for the connection to the Abelian sandpile model and Biggs' dollar game [4]). The graph parameter dgon(G) can be described in terms of a chip-firing game as follows (we will give a more formal definition later on). At any stage, we have a chip configuration consisting of a nonnegative number of chips at each vertex of G. We can move to a different chip configuration by firing a subset U ⊆ V (G) of vertices: for each edge uv with u ∈ U and v ∈ U we remove one chip from u and add one chip to v. Firing a subset U is only allowed if every vertex u ∈ U has at least as many chips as it has edges to vertices outside U (otherwise it would be left with a negative number of chips). A chip configuration is winning if for every vertex v there is a sequence of allowed moves that results in a configuration with at least one chip on v. The gonality dgon(G) of G is the smallest number of chips in a winning chip configuration.
Since the gonality of a graph is the sum of the gonalities of the connected components, we will restrict attention to connected graphs. It is well-known that the connected graphs of gonality 1 are precisely the trees. In fact it is easy to see that the gonality is invariant under adding leaves (vertices of degree 1). The graphs of gonality 2 were characterized by Chan [7]. Their result is stated in the more general context of metric graphs. Specialising to divisorial gonality on graphs, it implies Theorem (Implied by [7,Theorem 1.3]). Let G be a connected graph with no vertices of degree less than 2. Then dgon(G) = 2 if and only if there is a graph automorphism i : G → G of order 2 such that G/i is a tree.
No characterisation for graphs of gonality 3 is known. A trivial upper bound is dgon(G) |V |, which can be strengthened to dgon(G) |V | − α(G) for simple graphs, where α(G) denotes the stability number of G. An upper bound dgon(G) , matching the classical Brill-Noether bound, was conjectured by Baker [1,Conjecture 3.10].
In this paper we prove a lower bound for the gonality. Our main result is the following theorem that was conjectured in [11].
Theorem 1.1. Let G be a connected graph with treewidth tw(G) and gonality dgon(G). Then tw(G) dgon(G). Section 2 is devoted to preliminaries, including basic notation and terminology related to graphs, divisors and treewidth. In Section 3, we state and prove the main theorem. In Section 4, we consider some families of graphs for which treewidth equals gonality. These include: trees, grids, and complete multipartite graphs. In Section 5, we briefly review divisor theory for metric graphs. We show that the gonality of a metric graph is lower bounded by the gonality of a subdivision of the underlying graph. Hence, the treewidth is also a lower bound for metric graphs (tropical curves). In Section 6, we discuss some related notions of gonality defined in terms of harmonic morphisms, and show that there the treewidth is also a lower bound.

Preliminaries
The graphs in this paper will be finite and undirected (unless stated otherwise). We allow our graphs to have multiple (parallel) edges, but no loops. We will almost exclusively consider connected graphs. For a graph G, we denote by V (G) and E(G) the set of vertices and edges of G, respectively. By an edge uv, we mean an edge with ends u and v. For (not necessarily disjoint) subsets U, W ⊆ V (G), we denote by E(U, W ) the set of edges with an end in U and an end in W . For vertices u and v, we use the abbreviations The degree of a vertex v equals the number of edges with v as an endpoint and is denoted by is the graph with vertex set U and edge set consisting of the edges of G with both ends in U .
Let G = (V, E) be a connected graph. We can make G into an oriented graph by, for every edge e, assigning one end to be the head of e and the other end to be the tail of e. We view the edge e as oriented from its tail to its head. For a cycle C in G, we then denote by χ C ∈ R E the signed incidence vector defined by (1) χ C (e) = Similarly, we write χ P for the signed incidence vector of a path P . The incidence matrix M = M (G) ∈ R V ×E of G is defined by, for every v ∈ V and e ∈ E, setting The following two lemmas are well-known, and can be found in many books on algebraic graph theory (for example [13]). For the benefit of the reader, we will give the short proofs. (i) f is in the cut lattice of G, Proof. The implication from (iii) to (i) is trivial. The implication from (i) to (ii) follows since M χ C = 0 for every cycle C. For the implication from (ii) to (iii), let f ∈ Z E satisfy the condition in (ii). Let T be a spanning tree in G with root r, and Lemma 2.2. The null space of Q is spanned by the all-one vector 1.
Proof. Since the row sums of Q equal zero, it is clear that Q1 = 0. Conversely, let x be in the null space of Q and suppose, for contradiction, that x is not a multiple of 1. Since G is connected, we may choose v ∈ V for which x(v) is maximal and such that v has a neighbour u with x(u) < x(v). From Qx = 0 it follows that by our choice of v. This is a contradiction.
We denote by Div + (G) the set of effective divisors on G and by Div k We call two divisors D and D equivalent and write D ∼ D if there is an integer vector x ∈ Z V such that D − D = Q(G)x. Clearly, this is indeed an equivalence relation. Observe that equivalent divisors have equal degree as Q(G) has column sums equal to zero.
We will often consider the situation where x is the incidence vector of a subset U of V , that is, D = D − Q(G)1 U . Observe that in this case Algebraic Combinatorics, Vol. 3 #4 (2020) In terms of chip firing, we move one chip along each edge in the cut E(U, V U ). The following lemma shows that for equivalent effective divisors D and D , we can obtain D from D by a sequence of steps of this form such that each intermediate divisor is effective as well.
is effective for every t = 1, . . . , k and such that D k = D. Moreover, this chain is unique.
x is unique up to integral multiples of 1. Hence, there is a unique such x with the additional property that x 0 and supp( Uniqueness follows directly from the uniqueness of an The rank of a divisor D is defined as Observe that equivalent divisors have equal rank and that rank(D) deg(D).
Following Baker [1], we define the gonality of G by An effective divisor D is called v-reduced if for any nonempty subset U ⊆ V {v} the divisor D − Q(G)1 U is not effective (1) . In other words, for every nonempty Then there is a unique v-reduced divisor equivalent to D.
The set S is finite since the number of effective divisors equivalent to D is finite. The set S is nonempty as it contains the zero vector. Choose To show uniqueness, let D and D be two different, but equivalent effective divisors. It suffices to show that D and D are not both v-reduced. By Lemma 2.3 there are sets ∅ (1) The definition of v-reduced divisor can be extended to all divisors by allowing D(v) to be negative for a v-reduced divisor D as is done in [2]. However, in this paper we will mostly deal with effective divisors.
Algebraic Combinatorics, Vol. 3 #4 (2020) we have that C is a tree and |E({s}, V (C))| 1 for every s ∈ S. The following lemma is similar to a theorem of Luo [16] on rank determining sets in the context of metric graphs.
Lemma 2.6. Let S be a strong separator of G and let D be an effective divisor covering every s ∈ S. Then D has positive rank.
Proof. Since any superset of a strong separator is again a strong separator, we may assume that We have to show that S = V . Suppose not. Let C be a component of G[V S] and let S := {s ∈ S : |E({s}, V (C))| = 1}. Since G is connected, S is not empty, so we may take s ∈ S and assume that D is s-reduced. If S ⊆ supp(D), then D + Q(G)1 V (C) is effective and has support on at least one vertex in V (C) ⊆ V S, but this contradicts (7). Therefore, S ⊆ supp(D).
Choose t ∈ S supp(D). By (7), t is covered by D. However, t is not in the support of D, so in particular D is not t-reduced. Let a and b be the unique neighbours of s and t in V (C), respectively, and let P = (s, a, . . . , b, t) be the path from s to t with its interior points in V (C). Since D is s-reduced, but not t-reduced, there is a set U ⊆ V with s ∈ U , t ∈ U such that D := D − Q(G)1 U is effective. The cut E(U, V U ) must intersect some edge e = uv of the path P , and we find that D(u) 1 and D (v) 1. This contradicts (7), since at least one of u and v is in V (C) ⊆ V S.

2.2.
Treewidth. The notion of treewidth was first introduced by Halin [14] and later rediscovered by Robertson and Seymour [18] as part of their graph minor theory. There are several equivalent definitions of treewidth. We will follow Diestel [10], and define treewidth in terms of tree-decompositions.
Let G = (V, E) be a graph (we allow multiple edges and loops). Let (X i ) i∈I be a family of vertex sets X i ⊆ V indexed by the nodes of a tree T = (I, F ). The pair (T, (X i ) i∈I ) is a tree decomposition of G if it satisfies the following conditions: (ii) for every edge e = vw ∈ E there is an i ∈ I with v, w ∈ X i ; (iii) if i, j, k ∈ I and node j is on the path in T between nodes i and k, then The width of the tree decomposition is max i∈I |X i | − 1. The treewidth of a G is the minimum width of a tree decomposition of G.
In order to use treewidth as a lower bound, we will use a characterisation of treewidth by Seymour and Thomas [19] in terms of "brambles".  E(B, B ). In the latter case, we say that B and B touch. A set S ⊆ V is called a hitting set for B if it has nonempty intersection with every member of B. The order of B, denoted B , is the minimum size of a hitting set for B. That is: The following theorem by Seymour and Thomas [19] gives a min-max characterisation of treewidth. Remark 2.9. We note that the treewidth of a graph is equal to the treewidth of the underlying simple graph, as can be seen directly from the definition. It is wellknown that treewidth is monotone under taking minors (see for example [10]). That is, removing or contracting edges can only decrease treewidth. In particular, if H is a subdivision of G, then tw(G) tw(H).
We refer the interested reader to Chapter 12 in [10] for an excellent exposition of treewidth and its role in the graph minor theory.

Proof of the main theorem
In this section we prove our main theorem. We start by stating and proving two lemmas.
This is a contradiction as well, so we see that B ⊆ U must hold. Proof. We will construct a hitting set for B of size at most |E(U, V U )| + 1. Let F := E(U, V U ) be the cut determined by U and let H := (V, F ). Let be the "shores" of the cut F . Let B ∈ B be such that B ⊆ V U . Let B := {B ∈ B | B ⊆ U }. By assumption, B is nonempty. Choose B ∈ B for which B ∩ X is inclusion-wise minimal. Observe that B ∩ X is nonempty, since B must touch B.
(3) The article of Seymour and Thomas [19] used the term screen instead of bramble, but the latter has since become the standard.
Algebraic Combinatorics, Vol. 3 #4 (2020) We now define a hitting set S for B as follows. Add an arbitrary element s from B ∩ X to S. For each edge xy ∈ E(X, Y ) with x ∈ X, y ∈ Y , we add x to S if x ∈ B , and otherwise we add y to S. Hence |S| 1 + |F |. See Figure 1 for a depiction of the situation. Figure 1. The hitting set S for the bramble B is formed by the black vertices.
To prove that S covers B, consider any A ∈ B. First observe that A intersects X ∪Y .

In the first case G[A ∪ B] is not connected and in the second case G[A ∪ B ] is not connected. In both cases, this contradicts the fact that B is a bramble.
We consider the following three cases.
• Case A ∩ Y = ∅. In this case A ⊆ U . By the choice of B , we have either B ∩ X ⊆ A ∩ X and hence s ∈ A, or there exists an x ∈ (X ∩ A) B , which implies that x ∈ S. In both situations S intersects A.
be an edge e = xy with x ∈ B ∩ X and y ∈ A ∩ Y . By construction of S we have y ∈ S. Hence, S intersects A.
is connected, there is an edge e = xy with x ∈ X, y ∈ Y and x, y ∈ A. Since S contains at least one endpoint from each edge in F , the set S must intersect A. We conclude that S is a hitting set for B of size at most |E(U, V U )| + 1, which proves the lemma.
We now prove the main theorem. By definition of i, the set supp(D i−1 ) intersects every member of B that is intersected by supp(D). By our choice of D, the set supp(D) intersects at least as many members of B as supp(D i−1 ) does. Hence it follows that supp(D i−1 ) does not intersect B.
Since supp(D k ) does intersect B, there is an index j i such that B∩supp(D j−1 ) = ∅ and B ∩supp(D j ) = ∅. Hence, since

Examples
In this section, we describe some examples of (classes of) graphs for which equality holds in tw(G) dgon(G). To prove equality it suffices in each case to exhibit a positive rank divisor D and a bramble B such that B − 1 deg(D).     k-partite graph). Let G = (V, E) be a complete k-partite graph, k 2, with partition V = V 1 ∪ · · · ∪ V k , where n i := |V i | 1. We may assume that n 1 n 2 · · · n k .
For i = 1, . . . , k let s i ∈ V i and consider the bramble B := {{s 1 }, . . . , {s k }} ∪ {{u, v} | uv ∈ E}. A set S ⊆ V is a hitting set for B if and only if s 1 , . . . , s k ∈ S and there is at most one index i such that V i ⊆ S. Hence a hitting set of minimal cardinality is given by S := V 1 ∪ · · · ∪ V k−1 ∪ {s k }. Hence tw(G) B − 1 = n 1 + · · · + n k−1 .
. . , n. Hence, since every (a, b) ∈ V is in the support of some D b , the rank of D 1 is at least one. Therefore we can conclude that m + 1 tw(G) dgon(G) m + 1, and hence dgon(G) = tw(G) = m + 1.
An interesting family for which we do not know the answer is the following. Let Q n be the n-dimensional cube. That is, Q n is the graph with vertex set {0, 1} n and two vertices x, y are connected by an edge if x and y differ in exactly one coordinate. It is clear that dgon(Q n ) 2 n−1 and we believe that equality holds. On the other hand, tw(Q n ) = Θ( 2 n √ n ); see [8].

Metric graphs
In this section, we show that for any metric graph Γ (tropical curve) with underlying connected graph G, there is a subdivision H of G, such that dgon(H) dgon(Γ). Hence, the treewidth is also a lower bound for metric graphs: (11) tw(G) tw(H) dgon(H) dgon(Γ).
Let G = (V, E) be a connected graph, and let l : E → R >0 be a length function on the edges. Associated to the pair (G, l) is the metric graph Γ which is the compact connected metric space obtained by identifying every edge e with a real interval of length l(e) and glueing edges along common endpoints. The free abelian group on the points of Γ is denoted Div(Γ) and the elements of Div(Γ) are called divisors on Γ. For D = c 1 v 1 + · · · + c k v k ∈ Div(Γ), with c 1 , . . . , c k ∈ Z {0} and v 1 , . . . , v k ∈ Γ distinct, the degree of D is defined as deg(D) := c 1 + · · · + c k and the support of D is denoted supp(D) A rational function on Γ is a continuous real valued function on Γ such that its restriction to any edge is piecewise linear with finitely many pieces and integral slopes. The set of rational functions on Γ is denoted Rat(Γ). Let f ∈ Rat(Γ). For each v ∈ Γ, let c v be the sum of the outgoing slopes of f at v. So c v = 0 only for breakpoints of f (which may include points in V ). The associated divisor is denoted div(f ) := v∈Γ c v v and is called a principal divisor. The set of principal divisors is denoted Prin(Γ) and is a subgroup of Div(Γ). Two divisors are equivalent if their difference is a principal divisor. Observe that a principal divisor has degree 0, and hence equivalent divisors have equal degrees.
Algebraic Combinatorics, Vol. 3 #4 (2020) For a point v ∈ Γ, we say that an effective divisor D covers v if there exists an effective divisor equivalent to D with v in its support. The gonality dgon(Γ) is defined as the minimum degree of a divisor that covers every point v ∈ Γ. It was proven in [16] that if D covers every v ∈ V , then D covers every v ∈ Γ. However, we will not use that result here.
For any subset S ⊆ Γ, we denote Observe that Prin S (Γ) = Div S (Γ) ∩ Prin(Γ) and that two divisors D, D ∈ Div S (Γ) are equivalent if and only if D − D ∈ Prin S (Γ).
The elements of Div V (Γ) can be identified with the corresponding elements in Z V and thus viewed as the divisors on the graph G. Up to adding a constant function, the elements of Rat V (Γ) are determined by their integral slopes of the edges of G. To this end, we fix an arbitrary orientation on G, and define the map φ : Rat V (Γ) → Z E by setting φ(f )(e) to be the slope of f on edge e (in the forward direction). It is easy to see that g ∈ Z E is in the image of φ if and only if (15) e∈E g(e)l(e)χ C (e) = 0 for every cycle C in G.
Observe that for f ∈ Rat V (Γ) we have: div(f ) = −M φ(f ), where M is the signed vertex-edge incidence matrix of G.
In the case l = 1, if follows from (15)   Proof. Let D be a minimum degree effective divisor covering Γ. In particular, D covers every v ∈ V . Hence, for every v ∈ V , there is an effective divisor D v equivalent to D with v in its support. Let V := V ∪ supp(D) ∪ v∈V supp(D v ). Let Γ be obtained by subdividing Γ at the points in V V . Denote by G and l the corresponding underlying graph and length function so that Γ is the metric graph associated with (G , l ). The divisor D and the divisors D v can now be seen as equivalent elements of We equip G with an arbitrary orientation. It follows that y = l is a solution to the system (17) e∈G φ(f v )(e)y(e)χ C (e) = 0 for every cycle C in G and every v ∈ V .
Since (17) is a finite rational linear system in y, and since l > 0 is a solution, the system also has a solution l ∈ Z E >0 . It follows that the D v are equivalent divisors on the metric graph associated with (G , l ). Subdividing every edge e of G into l parts to obtain a graph H, we can view the D v as equivalent divisors in Div V (Γ ), where Γ is the metric graph associated to (H, 1) in which all edges have length one. Finally, this implies that the D v are also equivalent as divisors of H. It follows that Algebraic Combinatorics, Vol. 3 #4 (2020)