Tropical curves of hyperelliptic type

Abstract

We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends only on the underlying graph of a tropical curve and is preserved when passing to genus $\ge 2$ connected minors. The main result is an forbidden minors characterization of tropical curves of hyperelliptic type.

1 Introduction

Stable gonality, the smallest degree of a harmonic morphism to a tree, is a well-studied invariant of graphs and tropical curves. Like many invariants of tropical curves, it varies unpredictability when taking minors of the underlying graph. Tree width, a lower bound for stable gonality [6], does behave well with respect to taking graph minors. More precisely, the set of graphs of tree width at most k is minor closed; therefore, it admits a forbidden minors characterization. In this paper, we focus on the case of stable gonality 2, i.e., hyperelliptic tropical curves. The lower bound by tree width implies that such a tropical curve does not admit a $K_4$ minor. We describe a new collection of tropical curves, those of hyperelliptic type, that interpolates between hyperelliptic tropical curves and those of tree width 2, is minor closed (suitably defined), and admits a simple forbidden minors characterization.

The classical Torelli theorem asserts that 2 algebraic curves are isomorphic if and only if their Jacobians are isomorphic as polarized abelian varieties. It is well known that the tropical analogue of this theorem is not true, see [9, Section 6.4]. For example, varying the lengths of a separating pair of edges while preserving their sum produces tropical curves with isomorphic Jacobians. This phenomenon is presented in Fig. 1. As a consequence, it is possible for a nonhyperelliptic tropical curve to have a Jacobian isomorphic to that of a hyperelliptic tropical curve. A tropical curve with the same Jacobian as a hyperelliptic tropical curve is said to be of hyperelliptic type. The property of being hyperelliptic type is independent of the edge lengths and preserved when passing to connected genus $\ge 2$ minors. Our main theorem is a forbidden minors classification of these tropical curves.

Fig. 1

Two tropical curves with isomorphic Jacobians

Theorem 1.1

A tropical curve $\Gamma$ is of hyperelliptic type if and only if the underlying graph of $\Gamma$ does not have $K_4$ or $L_3$ as a minor.

Here, $K_4$ is the complete graph on 4 vertices and $L_3$ is the “loop of 3 loops,” and both graphs are displayed in Fig. 2. We prove a slightly stronger result, see Theorem 4.5.

Fig. 2

Graphs $K_4$ (left) and $L_3$ (right)

In Sect. 2, we review topics in tropical curves that we will need in this paper, including a discussion of C1-sets and 3-edge connectivization. A key tool used throughout the paper is the tropical Torelli theorem of Caporaso and Viviani [4]. This will be used in Sect. 3 to give a description of hyperelliptic type in terms of 3-edge connectivizations. We will also show that hyperelliptic type depends only on the underlying graph and that it is a minor closed property. Graphs that have no $K_4$ or $L_3$ minor admit a particular type of nested ear decompositions. These will be introduced in Sect. 4, which will then be used to prove the main theorem.

2 Preliminaries

2.1 Tropical curves

A weighted graph $\mathbf {G} = (G,\omega )$ is a finite connected graph G (possibly with loops or multiple edges) together with a function $\omega :V(G) \rightarrow \mathbb {Z}_{\ge 0}$ recording the weights of the vertices. Each edge $e\in E(G)$ is viewed as a pair of distinct half-edges. We write $e=vw$ to indicate that the endpoints of e are the vertices v and w. The valence of a vertex v is the number of half-edges incident to v, written ${{\,\mathrm{val}\,}}(v)$. In particular, a loop counts for 2 incidences. A vertex v is stable if

$$\begin{aligned} 2\,\omega (v) - 2 + {{\,\mathrm{val}\,}}(v) > 0, \end{aligned}$$

and a weighted graph is stable if each vertex is stable. The genus of $\mathbf {G}$ is

$$\begin{aligned} g(\mathbf {G}) = b_1(G) + |\omega | \end{aligned}$$

where $b_1$ is the first Betti number of G and $|\omega |$ is the sum of the weights. For an edge e of $\mathbf {G}$, the contraction of $\mathbf {G}$ by e is the weighted graph $\mathbf {G}/e$ obtained by contracting e while changing the weight function in the following way. If e is a loop edge incident to v, then the weight of v increases by 1. If e is an edge between distinct vertices $v_1$ and $v_2$, then the weight of the new vertex is $\omega (v_1) + \omega (v_2)$. Note that this preserves the genus and stability of $\mathbf {G}$. If $\mathbf {G}'$ is obtained from $\mathbf {G}$ by a sequence of contractions, then $\mathbf {G}'$ is a specialization of $\mathbf {G}$.

A tropical curve $\Gamma$ is a weighted graph $\mathbf {G}$ together with a function $\ell : E(G) \rightarrow \mathbb {R}_{>0}$ recording the length of each edge. Two tropical curves are tropically equivalent if one can be obtained from the other via a sequence of the following moves:

• adding or removing a 1-valent vertex v with $\omega (v) = 0$ and the edge incident to v, or

• adding or removing a 2-valent vertex v with $\omega (v) = 0$, preserving the underlying metric space.

Every tropical curve of genus $g\ge 2$ is tropically equivalent to a unique tropical curve whose underlying weighted graph is stable [3, Section 2]. We refer to this as the stable model for $\Gamma$.

Let $\Gamma$ be a genus $g \ge 2$ tropical curve. The Jacobian of $\Gamma$ is the real g-dimensional torus

$$\begin{aligned} {{\,\mathrm{Jac}\,}}(\Gamma ) = (H_1(\Gamma ,\mathbb {R}) \oplus \mathbb {R}^{|\omega |})/(H_1(\Gamma ,\mathbb {Z}) \oplus \mathbb {Z}^{|\omega |}) \end{aligned}$$

together with the semi-positive quadratic form $Q_{\Gamma }$ which vanishes on $\mathbb {R}^{|\omega |}$ and on $H_1(\Gamma ,\mathbb {R})$ is equal to

$$\begin{aligned} Q_{\Gamma } \left( \sum _{e\in E(\Gamma )} \alpha _{e}\cdot e \right) = \sum _{e\in E(\Gamma )} \alpha _e^2\cdot \ell (e). \end{aligned}$$

Note that ${{\,\mathrm{Jac}\,}}(\Gamma )$ is canonically identified with ${{\,\mathrm{Jac}\,}}(\Gamma ')$ when $\Gamma$ and $\Gamma '$ are tropically equivalent, see [9, Section 6.1]. For details on Jacobians in the weighted case, we refer the reader to [2, Definition 5.1.1].

2.2 Two-isomorphism of tropical curves

The cycle matroid of a weighted graph $\mathbf {G}$ is the cycle matroid of G with the addition of |w| loops. Two weighted graphs are 2-isomorphic if there is a bijection on the edge sets that induces an isomorphism of their cycle matroids. The 2-isomorphism class of $\mathbf {G}$ is written as $[\mathbf {G}]_2$. The following lemma, which is [4, Remark 2.2.6], is a consequence of the main theorem in [11].

Lemma 2.1

If G has no cut vertices or separating pairs of vertices, then $[G]_2 = \{G\}$.

Two tropical curves $\Gamma = (\mathbf {G},\ell )$ and $\Gamma ' = (\mathbf {G}', \ell ')$ of the same genus are 2-isomorphic if there is a length-preserving bijection on the edge sets that induces an isomorphism on the cycle matroids of $\mathbf {G}$ and $\mathbf {G}'$. The 2-isomorphism class of $\Gamma$ is written as $[\Gamma ]_2$.

2.3 Connectivity and C1-sets

Let $\mathbf {G}=(G,\omega )$ be a weighted graph. Then, $\mathbf {G}$ is 2-connected if $\omega (v) = 0$ for all v and G has no cut-vertices. Every weighted graph admits a decomposition into blocks, i.e., subgraphs that are either a single vertex of weight 1 or a maximal 2-connected subgraph of $\mathbf {G}$. See Fig. 3 for an example of such a decomposition.

Fig. 3

A decomposition of a weighted graph into blocks

Denote the set of nonseparating edges of $\mathbf {G}$ by $E(\mathbf {G})_{{{\,\mathrm{ns}\,}}}$. For $e,f\in E(\mathbf {G})_{{{\,\mathrm{ns}\,}}}$, we say that $e\sim f$ if $e=f$ or (e, f) form a separating pair of edges. This determines an equivalence relation on $E(\mathbf {G})_{{{\,\mathrm{ns}\,}}}$ (see [4, Lemma 2.3.2]) whose equivalence classes are called C1-sets. Write ${{\,\mathrm{Sets}\,}}^1(\mathbf {G})$ for the collection of C1-sets. The C1-set that contains $f\in E(\mathbf {G})_{{{\,\mathrm{ns}\,}}}$ is denoted by $S_{f}$.

The weighted graph $\mathbf {G}$ is k-edge connected if G has at least 2 edges, and the graph obtained by removing any $k-1$ edges from G is connected. The 2-edge connectivization of $\mathbf {G}$, written $\mathbf {G}^2$, is obtained by contracting all separating edges of $\mathbf {G}$. Consider the following operation on $\mathbf {G}$.

• (C) Given $S'\subset S$ for $S \in {{\,\mathrm{Sets}\,}}^1(\mathbf {G})$ and $e_0\in S'$, contract all edges in $S'$ except $e_0$.

Let $S_1,\ldots ,S_k$ be the C1-sets of $\mathbf {G}^2$ and fix $e_i\in S_i$. A 3-edge connectivization of $\mathbf {G}$ is a weighted graph $\mathbf {G}^3$ obtained by forming $\mathbf {G}^2$, then applying move (C) to each $(S_i,e_i)$ $i=1,\ldots ,k$. While a weighted graph may have many different 3-edge connectivizations, they all belong to the same 2-isomorphism class [4, Lemma 2.3.8]. By the same cited Lemma, the contraction map $\mathbf {G}\rightarrow \mathbf {G}^3$ induces a bijection between ${{\,\mathrm{Sets}\,}}^1(\mathbf {G})$ and $E(\mathbf {G}^3)$. Given S in ${{\,\mathrm{Sets}\,}}^1(\mathbf {G})$, let $e_S$ denote the edge under this correspondence. Then, we have a map $\psi :E(\mathbf {G})_{{{\,\mathrm{ns}\,}}} \rightarrow E(\mathbf {G}^3)$ given by sending f to $e_{S_f}$. Two weighted graphs $\mathbf {G}$ and $\mathbf {G}'$ are C1-equivalent, written $\mathbf {G}\sim _{C1} \mathbf {G}'$, if they belong to the same equivalence class of the equivalence relation generated by (C).

The C1-sets of a tropical curve $\Gamma = (\mathbf {G},\ell )$ are the C1-sets of $\mathbf {G}$, and the 2-edge connectivization $\Gamma ^2$ is obtained by contracting all separating edges of $\Gamma$. A 3-edge connectivization $\Gamma ^3$ of $\Gamma$ is formed in a manner similar to that of a weighted graph, but the edge lengths are modified so that ${{\,\mathrm{Sets}\,}}^1(\Gamma ) \rightarrow E(\Gamma ^3)$ is volume preserving. More precisely, consider the following operation on $\Gamma$.

(C$'$):

Given $S'\subset S$ for S in ${{\,\mathrm{Sets}\,}}^1(\Gamma )$ and $e_0\in S'$, contract all edges in $S'$ except $e_0$ and set the length of $e_0$ to $\sum _{e\in S'} \ell (e)$.

Let $S_1,\ldots ,S_k$ be the C1-sets of $\Gamma ^2$ and fix $e_i\in S_i$. A 3-edge connectivization of $\Gamma$ is a tropical curve $\Gamma ^3 = (\mathbf {G}^3,\ell ^3)$ obtained by first forming $\Gamma ^2$ and then by applying move (C’) to each $(S_i,e_i)$ $i=1,\ldots ,k$. Any 2 3-edge connectivizations of a tropical curve are 2-isomorphic [4, Remark 4.1.8]. We say that $\Gamma$ and $\Gamma '$ are C1-equivalent, written $\Gamma \sim _{C1} \Gamma '$, if $\Gamma ^2$ and $\Gamma '^2$ belong to the same equivalence class of the equivalence relation generated by move (C’).

Proposition 2.2

Let $\Gamma = (\mathbf {G},\ell )$ be a tropical curve and $\mathbf {G}'$ a weighted graph.

• 1. If $\Gamma '$ is a tropical curve such that $\Gamma \sim _{C1} \Gamma '$, then $[\Gamma ^3]_2 = [\Gamma '^3]_2$.

• 2. If $\mathbf {G}\sim _{C1} \mathbf {G}'$, then there exists a $\ell '$ such that $(\mathbf {G}, \ell ) \sim _{C1} (\mathbf {G}',\ell ')$.

• 3. If $\epsilon : \mathbf {G}'^3 \rightarrow \mathbf {G}^3$ is a 2-isomorphism, then there is a bijection $\beta :{{\,\mathrm{Sets}\,}}^1(\mathbf {G}') \rightarrow {{\,\mathrm{Sets}\,}}^1(\mathbf {G})$ so that $\beta (S_{e'}) = S_{\epsilon (e')}$.

Proof

Statements (1) and (3) are clear. For (2), it suffices to consider the case when $\mathbf {G}'$ is obtained from $\mathbf {G}$ by applying (C) to $(S,e_0)$ where S is a C1-set and $e_0\in S$. Define $\ell '(e_0) = \sum _{e\in S} \ell (e)$, and $\ell '(e) = \ell (e)$ for $e\in E(\mathbf {G}){\setminus } S$. Applying move (C’) to $(\mathbf {G},\ell )$ yields $(\mathbf {G}',\ell ')$, as required. $\square$

2.4 Hyperelliptic tropical curves

For a more comprehensive treatment of hyperelliptic tropical curves, we refer the reader to [5] in the unweighted case, or [1, Section 4.11] in the weighted case.

Let $\Gamma = (\mathbf {G},\ell )$ be a tropical curve. An involution of $\Gamma$ is an involution of the underlying graph of $\mathbf {G}$ that preserves the weight and length functions. Note that swapping the 2 half-edges of a loop is a nontrivial involution. If $\tau$ exchanges the 2 half-edges e, then e is said to be flipped. The quotient of $\Gamma$ by $\tau$ is the (unweighted) tropical curve $\Gamma /\tau$ whose vertices of $\Gamma /\tau$ correspond to orbits of the action of $\langle \tau \rangle \subset {{\,\mathrm{Aut}\,}}(\mathbf {G})$ on $V(\mathbf {G})$, and the edges of $\Gamma /\tau$ correspond to the orbits of the nonflipped edges of $\mathbf {G}$. The length of $[e] \in E(\Gamma /\tau )$ is $|{{\,\mathrm{Stab}\,}}(e)| \cdot \ell (e)$. In particular, a flipped edge is collapsed to a vertex upon forming $\Gamma /\tau$.

A tropical curve of genus at least 2 is hyperelliptic if its stable model $\Gamma$ has an involution $\tau$ such that each vertex of positive weight is fixed, and the underlying graph of $\Gamma /\tau$ is a tree. If $\Gamma$ is hyperelliptic, then there exists a unique $\tau$ that fixes all separating edges pointwise, see [5, Corollary 3.8]. This $\tau$ is called the hyperelliptic involution of $\Gamma$.

We end this section with a discussion of fixed points and C1-sets of a stable hyperelliptic tropical curve. Let $\Gamma$ be a stable hyperelliptic tropical curve, $\tau$ its hyperelliptic involution, $T = \Gamma /\tau$, and $\pi :\Gamma \rightarrow T$ the quotient map. A fixed point of $\tau$ is a vertex v of some subdivision of $\Gamma$ so that $\tau (v) = v$.

Lemma 2.3

Let a be a 1-valent vertex of T.

• 1. If $\pi ^{-1}(a)$ contains a single vertex v, then v is a fixed point.

• 2. If $\pi ^{-1}(a)$ contains exactly 2 distinct vertices $v,v'$, then there are at least 2 edges between v and $v'$, and their midpoints are fixed by $\tau$.

Proof

Suppose v is the only vertex in $\pi ^{-1}(a)$. If $\omega (v) > 0$, then v is fixed by definition. Let s be the unique edge in T adjacent to a. Since $\pi ^{-1}(s)$ contains at most 2 edges and $\Gamma$ is stable, there must be a loop edged based at v. So v is a cut vertex, which is a fixed point by [5, Lemma 3.9].

Now suppose $\pi ^{-1}(a)$ contains the distinct vertices $v,v'$. Because $\tau (v) = v'$, $\omega (v)$ and $\omega (v')$ are both 0. Moreover, $\pi ^{-1}(s)$ contains exactly 2 edges e and $e'$ where e, resp. $e'$, is adjacent to v, resp. $v'$. By stability, there must be at least 2 edges between v and $v'$. Since these edges are flipped, their midpoints are fixed by $\tau$. $\square$

Lemma 2.4

Let Q be a path in T from vertices a to b, and $v\in \pi ^{-1}(a)$. Then, Q lifts to a path P in $\Gamma$ starting at v such that $\pi (P) = Q$ and P contains no flipped edges.

Proof

Suppose Q is given by the sequence of consecutive edges $(s_1,\ldots , s_k)$ where $s_i = a_{i}a_{i+1}$ with $a_1=a$. We proceed by induction on k. If $k=0$, then P consists of only v. Now let $(e_{1},\ldots ,e_j)$ be a lift of $(s_1,\ldots , s_j)$ such that $e_1$ has v as an endpoint, each $e_i$ is a nonflipped edge, and $\pi (e_i) = s_i$. Choose an edge $e_{j+1}$ such that $\pi (e_{j+1}) = s_{j+1}$ (in particular $e_{j+1}$ is not flipped) and $e_{j+1}$ shares an endpoint with $e_j$. Then, $(e_{1},\ldots ,e_{j+1})$ is the requisite lift of $(s_1,\ldots ,s_j)$. $\square$

Proposition 2.5

Let S be a C1-set of $\Gamma$. Then, either

• S has 1 edge and $\tau$ flips it, or

• S has 2 edges and $\tau$ exchanges them.

Proof

Without loss of generality, assume $\Gamma$ is 2-edge connected. Let $e=uu'$ and $f=vv'$ be distinct edges. It suffices to show that (e, f) form a separating pair if and only if $\tau (e) = f$.

Case 1. Suppose e and f are both flipped. By symmetry, it suffices to find a path from v to $v'$ avoiding e and f. Let w be any fixed point of $\tau$ not contained in the interior of e or f; such a fixed point exists by Lemma 2.3. There are paths in T from $\pi (v)$, resp. $\pi (v')$, to $\pi (w)$. By Lemma 2.4, these lift to paths from v, resp. $v'$, to w that do not pass through flipped edges (except possibly one containing w), avoiding e and f.

Case 2. Suppose f is flipped but e is not. Let w be a fixed point lying above a 1-valent vertex of T in the component of $T {\setminus } \pi (e)$ containing $\pi (f)$. We may choose w so that it is not in the interior of f. There is a path in $T{\setminus } \pi (e)$ from $\pi (v) = \pi (v')$ to $\pi (w)$. By Lemma 2.4, this lifts to paths from v, resp. $v'$, to w passing through only nonflipped edges (except possibly w) while avoiding e, giving the requisite path from v and $v'$.

Now we construct a path from u to $u'$ in $\Gamma {\setminus }\{e,f\}$. In a manner similar to the previous paragraph, we can find paths from u to $\tau (u)$ and $u'$ to $\tau (u')$ avoiding e and f. Together with $\tau (e)$, this gives the requisite path.

Case 3. Suppose neither e or f are flipped. If $\tau (e) = f$, then (e, f) is a separating pair of edges. Now assume that $\tau (e) \ne f$. By symmetry, it suffices to construct a path from v to $v'$ avoiding e and f. If there are paths in T from $\pi (v)$ and $\pi (v')$ to a 1-valent vertex that avoid $\pi (e)$ and $\pi (f)$, then connecting v and $v'$ is similar to Case 2. Otherwise, every vertex a in the maximal subgraph between $\pi (e)$ and $\pi (f)$ is 2-valent. By stability, each $\pi ^{-1}(a)$ contains a fixed point, which can be used to construct the desired path from v to $v'$, as in the previous cases. $\square$

3 Hyperelliptic type and its properties

A tropical curve $\Gamma$ is said to be of hyperelliptic type if there is a hyperelliptic tropical curve $\Gamma '$ such that $({{\,\mathrm{Jac}\,}}(\Gamma ), Q_{\Gamma }) \cong ({{\,\mathrm{Jac}\,}}(\Gamma '), Q_{\Gamma '})$. Such a $\Gamma '$ is called a hyperelliptic model of $\Gamma$. Since hyperelliptic type is preserved under tropical equivalence, we are free to assume that the underlying weighted graph of a hyperelliptic type tropical curve is stable. Recall that the tropical Torelli theorem  [4, Theorem 4.1.9] (and [2, Theorem 5.3.3] in the vertex-weighted case) states that $({{\,\mathrm{Jac}\,}}(\Gamma ), Q_{\Gamma }) \cong ({{\,\mathrm{Jac}\,}}(\Gamma '), Q_{\Gamma '})$ if and only if $[\Gamma ^3]_2 = [\Gamma '^3]_2$. Therefore, we have the following characterization of hyperelliptic-type tropical curves.

Proposition 3.1

A genus $g\ge 2$ tropical curve $\Gamma$ is of hyperelliptic type if and only if there is a hyperelliptic tropical curve $\Gamma '$ such that $[\Gamma ^3]_2 = [\Gamma '^3]_2$.

By Propositions 2.2(1) and 3.1, we have the following Lemma.

Lemma 3.2

If $\Gamma$ is C1-equivalent to a tropical curve of hyperelliptic type, then $\Gamma$ is of hyperelliptic type.

We say that $\Gamma$ is strongly of hyperelliptic type if there is a choice of edge lengths that make it hyperelliptic. By Proposition 3.3, being of hyperelliptic type does not depend on the edge lengths. Therefore, strongly hyperelliptic-type tropical curves are of hyperelliptic type. However, the converse is not true. Consider the tropical curves in Fig. 4. The one on the left is hyperelliptic type (a hyperelliptic model is displayed on the right), but no choice of edge lengths will make it hyperelliptic. It is not even a specialization of a hyperelliptic tropical curve.

Fig. 4

On the left is a tropical curve of hyperelliptic type that is not strongly of hyperelliptic type. On the right is a hyperelliptic model for this tropical curve

Now we show that the property of being hyperelliptic type does not depend on the length function.

Proposition 3.3

Suppose $\Gamma _1 = (\mathbf {G}, \ell _1)$ is of hyperelliptic type, and let $\Gamma _2 = (\mathbf {G}, \ell _2)$ be a tropical curve with the same underlying weighted graph. Then, $\Gamma _2$ is also of hyperelliptic type.

Proof

Without loss of generality, we may assume that $\mathbf {G}$ is 2-edge connected. Suppose $\Gamma _1' = (\mathbf {G}', \ell _1')$ is a stable hyperelliptic model for $\Gamma _1$. Let $\beta :{{\,\mathrm{Sets}\,}}^1(\mathbf {G}') \rightarrow {{\,\mathrm{Sets}\,}}^1(\mathbf {G})$ be the bijection of C1-sets as in Proposition 2.2(3). Using the characterization of C1-sets of $\mathbf {G}'$ as in Proposition 2.5, we define $\ell _2'$ on $\mathbf {G}'$ in the following way.

• If $S=\{e_0\}$ set $\ell _2'(e_0) = \sum _{e\in \beta (S)} \ell _2(e)$.

• If $S=\{e_0, f_0\}$ set $\ell _2'(e_0) = \ell _2'(f_0) = \frac{1}{2} \sum _{e\in \beta (S)} \ell _2(e)$.

The hyperelliptic involution on $\Gamma _1'$ induces one on $\Gamma _2' = (\mathbf {G}',\ell _2')$, hence $\Gamma _2'$ is hyperelliptic and $\Gamma _2 \sim _{C1} \Gamma _2'$. By Lemma 3.2, $\Gamma _2$ is of hyperelliptic type. $\square$

With this proposition in mind, we say that a weighted graph $\mathbf {G}$ is of hyperelliptic type if $(\mathbf {G},\ell )$ is hyperelliptic type for some (and therefore, any) length function $\ell$. Similarly, we say that $\mathbf {G}$ is strongly of hyperelliptic type if $(\mathbf {G},\ell )$ is hyperelliptic for some $\ell$.

Proposition 3.4

If $\Gamma = (\mathbf {G},\ell )$ is strongly of hyperelliptic type, then it is C1-equivalent to a hyperelliptic tropical curve.

Proof

Choose $\ell '$ so that $\Gamma ' = (\mathbf {G},\ell ')$ is a hyperelliptic tropical curve, and let $\tau$ be the graph involution of $\mathbf {G}$ induced by the hyperelliptic involution of $\Gamma '$. For each $e\in E(\Gamma )_{{{\,\mathrm{ns}\,}}}$, let $\ell ''(e) = (\sum _{f\in S_e} \ell (f))/|S_e|$. If e is a separating edge, let $\ell ''(e) = \ell (e)$. Set $\Gamma '' = (\mathbf {G},\ell '')$ and note that $\Gamma '' \sim _{C1} \Gamma$. If $S_e = \{e\}$, then $\tau$ flips e. Otherwise, $S_e = \{e,f\}$, and $\tau (e) = f$ by Proposition 2.5. Because $\ell ''(e) = \ell ''(f)$ for every C1-set of this form, $\tau$ induces a hyperelliptic involution on $\Gamma ''$. $\square$

In Theorem 4.5, we will prove that a tropical curve is of hyperelliptic type if and only if it is C1-equivalent to a hyperelliptic tropical curve.

In order for the wedge sum of 2 hyperelliptic tropical curves to be hyperelliptic, they need to be attached at fixed points of the respective hyperelliptic involutions. As we will see in the next 2 lemmas, such a wedge sum is of hyperelliptic type regardless of how we decide to glue. Given tropical curves $\Gamma _1 = (\mathbf {G}_1,\ell _1)$, $\Gamma _2 = (\mathbf {G}_2, \ell _2)$ and vertices $v_1$, $v_2$ of some subdivision of $\mathbf {G}_1$, $\mathbf {G}_2$, respectively, let $\Gamma _1\vee _{v_1,v_2} \Gamma _2$ denote the tropical curve obtained by gluing $\Gamma _1$ and $\Gamma _2$ at $v_1$ and $v_2$.

Lemma 3.5

Let $\Gamma _1$ and $\Gamma _2$ be weighted metric graphs, and $v_i, v_i'$ vertices of some subdivision of $\Gamma _i$. Set $\Gamma = \Gamma _1 \vee _{v_1, v_2} \Gamma _2$ and $\Gamma ' = \Gamma _1 \vee _{v_1', v_2'} \Gamma _2$. Then, $\Gamma \sim _{C1} \Gamma '$. In particular, any tropical curve $\Gamma$ is C1-equivalent to an arbitrary wedge sum of its blocks.

Proof

By symmetry, it suffices to consider the case where, e.g., $v_2=v_2'$. Suppose $v_1$ is connected to $v_1'$ by an edge e. Subdivide e into 2 edges, and let w be the new vertex. Then, $\Gamma$ and $\Gamma '$ are C1-equivalent to $\Gamma _1\vee _{w,v_2}\Gamma _2$. The lemma now follows by induction on the number of edges in a path from $v_1$ to $v_1'$. $\square$

Lemma 3.6

Suppose that each block of $\Gamma ^2$ of genus $\ge 2$ is C1-equivalent to a strongly hyperelliptic type tropical curve. Then, $\Gamma$ is C1-equivalent to a hyperelliptic tropical curve. In particular, $\Gamma$ is of hyperelliptic type.

Proof

Let $\{\Gamma _i\}_{i=1}^k$ be the blocks of $\Gamma ^2$, and let $G_i$ be the underlying (unweighted) graph of $\Gamma _i$. If $g(G_i) \ge 2$, let $\Gamma _i'$ be a hyperelliptic tropical curve that is C1-equivalent to $\Gamma _i$ and $v_i$ a fixed point for its hyperelliptic involution. Note that $\Gamma _i'$ exists by Proposition 3.4, and fixed points exist by Lemma 2.3. If $g(G_i) = 1$, let $\Gamma _i'$ be the graph with a single vertex $v_i$ and a loop whose length is the sum of edge lengths of $\Gamma _i$. Finally, if $g(G_i) = 0$, then $\Gamma _i$ consists of a single vertex $v_i$ of weight 1. In this case, set $\Gamma _i' = \Gamma _i$. Then, $\vee _{v_i} \Gamma _i'$ is a hyperelliptic tropical curve that is C1-equivalent to $\Gamma$ by Lemma 3.5. The last statement follows from Lemma 3.2. $\square$

If $e\in E(\Gamma )_{{{\,\mathrm{ns}\,}}}$, then $\Gamma ^3{\setminus } \psi (e)$ may not be a 3-edge connectivization of $\Gamma {\setminus } e$. However, both of these tropical curves will have a common 3-edge connectivization, as we will see in the following Lemma.

Lemma 3.7

Given a tropical curve $\Gamma$ and $e \in E(\Gamma )_{{{\,\mathrm{ns}\,}}}$, $(\Gamma ^3 {\setminus } \psi (e))^3$ is a 3-edge connectivization of $\Gamma {\setminus } e$.

Proof

First, note that $\psi :E(\Gamma )_{{{\,\mathrm{ns}\,}}} \rightarrow E(\Gamma ^3)$ induces a map $\gamma : {{\,\mathrm{Sets}\,}}^1(\Gamma {\setminus } e) \rightarrow {{\,\mathrm{Sets}\,}}^1(\Gamma ^3{\setminus } \psi (e))$ by $\gamma (S) = \{ \psi (f) \, | \, f\in S \}$. Let S be a C1-set of $\Gamma {\setminus } e$ and write

$$\begin{aligned} \gamma (S)&= \{g_1,\ldots ,g_{k}\},S = \{f_{ij} \, | \, 1\le i\\&\quad \le k, 1 \le j \le |\psi ^{-1}(g_i)|\}, \end{aligned}$$

where $\psi (f_{ij}) = g_i$. The lemma now follows from the fact that applying move (C’) to $(S,f_{11})$ is the same as applying (C’) to each $(\psi ^{-1}(g_i),f_{i1})$, then to $(\gamma (S), g_1)$. $\square$

Now we will show that the property of being hyperelliptic type is a minor closed condition on stable weighted graphs. A minor of $\mathbf {G}$ is a weighted graph $\mathbf {G}'$ obtained by a sequence of deducting weights, removing edges or performing weighted contractions.

Proposition 3.8

Suppose $\mathbf {G}$ is a weighted stable graph of hyperelliptic type, and $\mathbf {G}'$ is a genus $g\ge 2$ connected minor. Then, $\mathbf {G}'$ is also of hyperelliptic type.

Proof

First, consider the case where $\mathbf {G}$ is strongly of hyperelliptic type, say $\Gamma = (\mathbf {G},\ell )$ is hyperelliptic and $\tau$ its involution. If $\Gamma '$ is obtained by deducting a weight by 1, then $\tau$ is still a hyperelliptic involution for $\Gamma '$. If $e \in E(\mathbf {G})_{{{\,\mathrm{ns}\,}}}$ is not in a separating pair, then $\tau$ flips it by Proposition 2.5. This means that $\Gamma /e$ and $\Gamma {\setminus } e$ remain hyperelliptic.

Now suppose e is in a pair of separating edges (e, f). Define a new length function $\ell '$ that agrees with $\ell$ except $\ell '(e) = \ell '(f) = \ell (f)/2$. Observe that $(\mathbf {G},\ell ')$ is hyperelliptic, and applying move (C’) to $(\{e,f\},f)$ in $(\mathbf {G},\ell ')$ yields $\Gamma /e$. Therefore, $(\mathbf {G},\ell ')$ is a hyperelliptic model of $\Gamma /e$ by Propositions 2.2(1) and 3.1. Finally, consider $\Gamma ' = \Gamma {\setminus } e$. Then, f is a separating edge of $\Gamma '$, say $\Gamma _1$ and $\Gamma _2$ are the components of $\Gamma ' {\setminus } f$. The map $\tau$ restricts to an involution on both $\Gamma _1$ and $\Gamma _2$ such that $\Gamma _i/\tau$ is a tree. By Lemma 3.6, $\Gamma '$ is of hyperelliptic type.

For the general case, fix a length function $\ell$ for $\mathbf {G}$ and let $\Gamma ' = (\mathbf {G}',\ell ')$ be a hyperelliptic model for $\Gamma = (\mathbf {G},\ell )$. As 3-edge connectivization and 2-isomorphism are independent of the weights, deducting a weight from $\Gamma$ produces a tropical curve that is of hyperelliptic type (as long as the genus remains at least 2). Fix a nonseparating edge e of $\mathbf {G}$. Suppose $\epsilon : E(\mathbf {G}^3) \rightarrow E(\mathbf {G}'^3)$ is a 2-isomorphism, and let $\psi :E(\Gamma )_{{{\,\mathrm{ns}\,}}} \rightarrow E(\Gamma ^3)$, $\psi ':E(\Gamma ')_{{{\,\mathrm{ns}\,}}} \rightarrow E(\Gamma '^3)$ be 3-edge connectivizations. Choose $e'\in E(\Gamma ')_{{{\,\mathrm{ns}\,}}}$ such that $\epsilon (\psi (e)) = \psi '(e')$. Then

$$\begin{aligned}{}[\Gamma ^3{\setminus } \psi (e)]_2 = [\Gamma '^3{\setminus } \psi '(e')]_2&[\Gamma ^3 / \psi (e)]_2 = [\Gamma '^3 / \psi '(e')]_2 \end{aligned}$$

and therefore

$$\begin{aligned}{}[(\Gamma ^3{\setminus } \psi (e))^3]_2 = [(\Gamma '^3{\setminus } \psi '(e'))^3]_2&[(\Gamma ^3 / \psi (e))^3]_2 = [(\Gamma '^3 / \psi '(e'))^3]_2. \end{aligned}$$

By applying Lemma 3.7 in the edge removal case, we see that

$$\begin{aligned}{}[(\Gamma {\setminus } e)^3]_2 = [(\Gamma '{\setminus } e')^3]_2&[(\Gamma /e)^3]_2 = [(\Gamma '/ e')^3]_2. \end{aligned}$$

The proposition now follows from the strongly hyperelliptic type case. $\square$

To a stable weighted graph $\mathbf {G}$, let $d(\mathbf {G})$ be

$$\begin{aligned} d(\mathbf {G}) = \sum _{v\in V(\mathbf {G})} ({{\,\mathrm{val}\,}}(v) + 3w(v) - 3). \end{aligned}$$

(3.9)

Note that $d(\mathbf {G})$ is the dimension of the stratum of $\overline{\mathcal {M}}_g$ of stable curves whose weighted dual graph is $\mathbf {G}$. Note that $d(\mathbf {G}) \ge 0$ with equality if and only if $w(v) = 0$ for all v and G is trivalent. For a tropical curve $\Gamma$, let $d(\Gamma ) = d(\mathbf {G})$ where $(\mathbf {G},\ell )$ is a stable tropical curve tropically equivalent to $\Gamma$. If $\mathbf {G}'$ is obtained from $\mathbf {G}$ by contracting an edge, then $d(\mathbf {G}') = d(\mathbf {G}) + 1$. In particular, if $d(\mathbf {G}^3) = d(\mathbf {G})$, then $\mathbf {G}$ is 3-edge connected.

Proposition 3.9

The graphs $K_4$ and $L_3$ are not of hyperelliptic type.

Proof

First, consider the $K_4$ case. Suppose $K_4$ is of hyperelliptic type, and let $\Gamma = (\mathbf {G}, \ell )$ be a stable hyperelliptic model for $K_4$. By the above comments, we see that

$$\begin{aligned} 0 \le d(\Gamma ) \le d(\Gamma ^3) = d(K_4) = 0, \end{aligned}$$

so $\mathbf {G}$ is already 3-edge connected. This means that $[\mathbf {G}]_2 = [K_4]_2$, and therefore $\mathbf {G}= K_4$ since $K_4$ is 3-vertex connected and Lemma 2.1. Because $K_4$ has no separating pairs of edges, the hyperelliptic involution of $\Gamma$ flips each edge of $K_4$ as in Proposition 2.5. This is a contradiction since no automorphism of $K_4$ satisfies this property.

Now suppose that $L_3$ is of hyperelliptic type and that $\Gamma =(\mathbf {G},\ell )$ is a stable hyperelliptic model. Any stable graph differing from $L_3$ by a single-edge contraction is 3-edge connected, so $[\mathbf {G}]_2 = [L_3]_2$. Because $L_3$ has no cut vertices or separating pairs of vertices, $[L_3]_2$ consists of just $L_3$, so $\mathbf {G}= L_3$. As in the $K_4$ case, this means that the hyperelliptic involution of $\Gamma$ flips each edge of $L_3$, which is a contradiction. $\square$

Remark 3.10

Observe that the “only if” direction of the theorem in Introduction follows from Propositions 3.8 and 3.9.

4 Nested ear decompositions

Nested ear decompositions were introduced in [8] to study series–parallel graphs. Since these graphs are characterized by the absence of a $K_4$ minor, it is natural to explore the link between nested ear decompositions and tropical curves of hyperelliptic type. In this section, we will introduce the notion of a hyperelliptic-type adapted (nested) ear decomposition and show that any graph with no $K_4$ or $L_3$ minor admits such a decomposition. This will then be used to prove the “if” direction of the theorem in Introduction.

Let G be a finite connected graph. An ear decomposition of G is a collection of paths $\mathcal {E}= \{E_0, \ldots , E_g\}$ called ears that partition E(G) and satisfy the following properties.

• 1. If 2 vertices in an ear are the same, then they must be the 2 endpoints of the same ear.

• 2. The 2 endpoints of $E_k$ ($k\ge 1$) appear in $E_i$ and $E_j$ for $i,j < k$.

• 3. No interior vertex of $E_j$ is in $E_i$ for $i< j$.

An ear decomposition $\mathcal {E}= \{E_0,\ldots , E_g\}$ is open if the 2 endpoints of each ear are distinct. The ear $E_j$ is nested in $E_i$ if $i<j$ and the endpoints of $E_j$ are in $E_i$. In this case, the nest interval of $E_j$ in $E_i$ is the path $E_{ij}$ in $E_i$ between the endpoints of $E_j$. We write $E_{ij}^{\circ }$ for the interior of $E_{ij}$. An ear decomposition $\mathcal {E}$ is nested if it is open and satisfies the following.

• 1. For $1\le j \le g$, there is some $i<j$ such that $E_j$ is nested in $E_i$.

• 2. If $E_j$ and $E_k$ are both nested in $E_i$, then either $E_{ij}$ and $E_{ik}$ have disjoint interiors, or one is contained in the other.

Let G be a graph with a nested ear decomposition $\mathcal {E}$. The ear $E_j$ is properly nested in $E_i$ if $E_j$ has an endpoint in the interior of $E_i$ and the other endpoint does not lie in the interior of any ear $E_k$ for $k>i$. If no such $E_i$ exists, i.e. the endpoints of $E_j$ coincide with those of $E_0$, then $E_j$ is called an initial ear. By [8, Lemma 3], every noninitial ear is properly nested in exactly one other ear. If $E_j$ is properly nested in $E_i$, we write $E_i \lessdot E_j$. Taking the reflexive and transitive closure of $\lessdot$ induces a partial order $\le$ on $\mathcal {E}$.

Proposition 4.1

A 2-connected graph G admits a nested ear decomposition $\mathcal {E}$ if and only if G has no $K_4$ minor.

Proof

By [7, Theorems 1, 3], G has no $K_4$ minor if and only if it is series–parallel. By [8, Theorem 1], G is series–parallel if and only if G has a nested ear decomposition. $\square$

A hyperelliptic-type adapted ear decomposition (HTED) is a nested ear decomposition $\mathcal {E}$ that satisfies the following additional property: If $E_{j}$ and $E_k$ are properly nested in $E_{i}$, then $E_{ij} \subset E_{ik}$ or $E_{ik} \subset E_{ij}$. A hyperelliptic adapted ear decomposition (HED) is a HTED satisfying the following.

• 1. If $E_i \lessdot E_j$, then the endpoints of $E_j$ lie in the interior of $E_i$.

• 2. If $E_{j}$, $E_{k}$ are nested in $E_i$ and $E_{ij} \subset E_{ik}$, then $E_{ij} = E_{ik}$ or the endpoints of $E_j$ lie in $E_{ik}^{\circ }$.

Lemma 4.2

If G is a 2-connected graph of genus $\ge 2$ that has no $K_4$ or $L_3$ minor, then G has a HTED.

Proof

By Proposition 4.1, G has a nested ear decomposition $\mathcal {E}= \{E_0,\ldots ,E_g\}$. By 2-connectedness, there are at least 2 initial ears $E_0$ and $E_1$. Label their endpoints by s and t. Suppose $\mathcal {E}$ is not hyperelliptic type adapted, i.e., there is an ear $E_i$ with at least 2 ears $E_j$ and $E_k$ properly nested in it whose nest intervals have disjoint interiors. Assume these are chosen maximally in the sense that if $E_i \lessdot E_{\ell }$, then $E_{i\ell }$ does not contain exactly one of $E_{ij}$ or $E_{ik}$. Without loss of generality, suppose that $E_1 \le E_i$. We claim that $\mathcal {E}$ consists of the following.

• 1. It has exactly 2 initial ears $E_0$ and $E_1$.

• 2. The only ear nested in $E_0$ is $E_1$.

• 3. The ears $E_j$ and $E_k$ are nested in $E_1$, and their nest intervals have disjoint interiors (take $E_j$ to be the ear closer to s along $E_1$).

• 4. If $E_m$ is nested in $E_1$, then $E_{1m}$ is contained in either $E_{1j}$ or $E_{1k}$.

• 5. If $E_{m}$ and $E_n$ are nested in $E_1$ with $E_{1m}$ and $E_{1n}$ contained in $E_{1j}$ (resp. $E_{1k}$), then $E_{1m}$ contains $E_{1n}$ or vice versa.

• 6. For every $E_{\ell } \ne E_{0}, E_{1}$, if $E_{m}$ and $E_{n}$ are nested in $E_{\ell }$, then $E_{\ell m}$ contains $E_{\ell n}$ or vice versa.

Fig. 5

Configurations of ears that have $L_3$ minors

If $E_{\ell } \ne E_1$ is nested in $E_0$ (possibly an initial ear), then any connected subgraph that contains $E_0$, $E_i$, $E_j$, $E_k$, and $E_{\ell }$ has a $L_3$ minor. Therefore, the only ear nested in $E_0$ is $E_1$. This proves (1) and (2). If $E_{i} \ne E_1$, then any connected subgraph containing $E_0$, $E_1$, $E_i$, $E_j$, and $E_k$ has a $L_3$ minor. Therefore, $E_i = E_1$, demonstrating (3). For illustrations of these cases, see the left graph in Fig. 5.

Suppose $E_{m}$ is nested in $E_{1}$. If $E_{1m}$ contains $E_{1j}$ and $E_{1k}$ then $E_{0} \cup E_{1} \cup E_j \cup E_k \cup E_m$ has a $L_3$ minor. The same is true if $E_{1m}^{\circ }$ is disjoint from $E_{1j}^{\circ }$ and $E_{1k}^{\circ }$, see the middle graph in Fig. 5. Together with the maximality assumption on $E_j$ and $E_k$, this proves (4). Now suppose $E_m$ and $E_n$ are nested in $E_{1}$ with $E_{1m}, E_{1n} \subset E_{1j}$. If $E_{1m}^{\circ }$ and $E_{1n}^{\circ }$ are disjoint, then $E_{0} \cup E_{1} \cup E_{k} \cup E_{m} \cup E_{n}$ contains a $L_3$ minor. Similarly, if $E_{1m}, E_{1n} \subset E_{1k}$, then they cannot have disjoint interiors, proving (5). For (6), suppose $E_{\ell } \ne E_{0}, E_1$ and that $E_m$, $E_n$ are nested in it. If $E_{\ell m}^{\circ }$ and $E_{\ell n}^{\circ }$ are disjoint, then any connected subgraph containing $E_{0}$, $E_{1}$, $E_{\ell }$, $E_{m}$, $E_n$ has a $L_3$ minor, see the right graph in Fig. 5. Thus, $\mathcal {E}$ has the required form.

Define a new ear decomposition $\mathcal {E}'$ in the following way. Write $t'$ for the endpoint of $E_k$ which is closer to s along $E_1$. Let $E_0'$ be the path from s to t along $E_0$, followed by the path from t to $t'$ along $E_1$. The next ear $E_1'$ will be the path from s to $t'$ along $E_1$. The remaining ears are left unchanged, i.e., $E_i'=E_i$ for $i\ge 2$. Then, $\mathcal {E}'$ is a HTED. For an illustration of this modification, see Fig. 6. $\square$

Fig. 6

Modification of a nested ear decomposition to a HTED on a graph with no $L_3$ minor

Lemma 4.3

Suppose G is a 2-connected stable graph of genus $g\ge 2$ that has a HTED.

• 1. The graph G has a HTED with at least 3 initial ears.

• 2. If G is trivalent, then any HTED with 3 initial ears is a HED.

• 3. There is a $G'$ that has a HED such that $G\sim _{C1} G'$ and G is a specialization of $G'$.

Proof

Suppose $\mathcal {E}= \{E_0, \ldots , E_g\}$ is a HTED for G that only has 2 initial ears $E_0$ and $E_1$. Let s, t be the endpoints of these ears. By stability, there is an ear $E_{j}$ ($j\ge 2$) that has s as an endpoint; assume that it is nested in $E_0$. Choose $E_j$ maximally in the sense that if $E_0 \lessdot E_k$, then $E_{0j}$ contains $E_{0k}$, and write $t'$ for the other endpoint of $E_j$. Define a new ear decomposition $\mathcal {E}'$ as follows. Let $E_0'$ be the path from s to $t'$ along $E_0$ and $E_1'$ the path from s to t along $E_1$, followed by the path from t to $t'$ along $E_0$. Finally, let $E_i'=E_i$ for $i\ge 2$. For an illustration of this modification, see Fig. 7. Then, $\mathcal {E}'$ is a HTED such that $E_0', E_1', E_j'$ are initial ears. This proves (1).

Fig. 7

Modification of a HTED to 1 that has 3 initial ears

Now suppose G is trivalent and $\mathcal {E}$ is a HTED with 3 initial ears. We claim that $\mathcal {E}$ is already a HED. Suppose $E_j \lessdot E_k$. If $E_j$ is initial, then both endpoints of $E_k$ lie in the interior of $E_j$ since $E_k$ is not initial. Now suppose $E_j$ is not initial and that a, b are the endpoints of $E_j$. Then, a lies in the interior of some ear $E_i$. Since G is trivalent, $E_k$ cannot have a as an endpoint. A similar argument shows that b is not an endpoint of $E_k$. Therefore, the endpoints of $E_k$ lie in the interior of $E_j$, and so condition (1) of a HED is satisfied.

Next, assume that $E_{j}$ and $E_{k}$ are nested in $E_i$ with $E_{ij} \subset E_{ik}$. If $E_{k}$ is initial, then either $E_{j}$ is initial (in which case $E_{i}$, $E_{j}$, and $E_k$ are the initial ears) or the endpoints of $E_j$ lie in $E_{ik}^{\circ }$. Otherwise, the endpoints of $E_k$ lie in the interior of $E_{i}$. Similar to the case in the previous paragraph, the endpoints of $E_j$ lie in $E_{ik}^{\circ }$. This verifies condition (2) of a HED and completes part (2) of this lemma.

To prove (3), we proceed by induction on d(G) defined in Eq. (3.9). When $d(G) = 0$, G is trivalent and $\mathcal {E}$ is already a HED. Now suppose that the lemma is true for stable graphs with $d<\delta$, and let G be a stable graph with $d(G) = \delta$. If $\mathcal {E}$ is not a HED, then there are ears satisfying at least one of the following:

• (a) $E_i\lessdot E_j$, but 1 endpoint of $E_j$ coincides with an endpoint a of $E_i$, or

• (b) $E_{ij} \subset E_{ik}$ but exactly 1 endpoint b of $E_j$ lies in the interior of $E_{ik}$.

Consider case (a). Assume $E_{j}$ is chosen so that if $E_i \lessdot E_{k}$, then $E_{ik} \subset E_{ij}$. Let $G'$ be the graph obtained from G in the following way. First, subdivide the edge in $E_{ij}$ adjacent to a, creating a new vertex b. Let $\mathcal {E}_a \subset \mathcal {E}$ be the ears $E_{\ell }$ that have a as an endpoint, and either $E_j\le E_{\ell }$ or $E_{i\ell } \subset E_{ij}$. For every ear in $\mathcal {E}_a$, move the corresponding endpoint from a to b. See the left side of Fig. 8 for an illustration. Let e be the unique edge in $E_i {\setminus } E_{ij}$ and f the edge between a and b. Then, (e, f) form a separating pair of edges for $G'$, and contracting f yields G. Therefore, $G'\sim _{C1}G$ and $d(G') = \delta - 1$. By the inductive hypothesis, there is a $G''$ that has a HED such that $G'' \sim _{C1} G'$ and G is a specialization of $G''$. Case (b) is handled in a similar fashion, see the right side of Fig. 8. $\square$

Fig. 8

Modifications in the inductive step of the proof of Lemma 4.3(3)

Lemma 4.4

Suppose G is a 2-connected stable graph of genus $g\ge 2$ that has a HED. Then, G is strongly of hyperelliptic type.

Proof

Fix a HED $\mathcal {E}= \{E_0,\ldots ,E_g\}$ for G. We define $\tau$ ear by ear as follows. Let v, w be the endpoints for $E_i$ and set $\tau (v) = w$. If $E_i$ has no ears properly nested in it, then $E_i$ has exactly 1 edge. Let $\tau$ flip this edge. Otherwise, choose a maximal collection of ears $E_{j_1},\ldots , E_{j_k}$ properly nested in $E_i$ so that $E_{ij_1} \subsetneq \cdots \subsetneq E_{ij_k} \subsetneq E_i$.

Now, $E_i{\setminus } E_{ij_1}$ separates the endpoints of each $E_{ij_a}$. Let $v_a$ be the endpoint of $E_{ij_a}$ appearing on the side of $E_i {\setminus } E_{ij_1}$ that contains v, and $w_a$ the other endpoint. Let $e_{i,a} = v_av_{a+1}$, $e_{i,k} = v_kv$, $f_{i,a} = w_{a+1}w_a$, and $f_{i,k} = ww_k$. Define $\tau (e_{i,a}) = f_{i,a}$ and $\tau (f_{i,a}) = e_{i,a}$. Finally, $E_{ij_1}$ consists of just 1 edge, so let $\tau$ flip it. This defines an involution $\tau$. Let $\pi : G \rightarrow G/\tau$ denote the quotient map.

We claim that $G/\tau$ is a tree. Suppose e is an edge of $G/\tau$. Then, $\pi ^{-1}(e)$ is an edge of G that is not flipped. This means that $\pi ^{-1}(e) = \{e_{i,a}, f_{i,a}\}$ for some i and a, which is a separating pair of edges. So removal of e from $G/\tau$ disconnects it. Since ever edge of $G/\tau$ is separating, it is a tree. Choosing $\ell$ so that $\ell (e_{i,a}) = \ell (f_{i,a})$ produces a hyperelliptic tropical curve $(G,\ell )$. $\square$

Theorem 4.5

Let $\Gamma = (\mathbf {G},\ell )$ be a tropical curve. The following are equivalent.

• 1. The tropical curve $\Gamma$ is of hyperelliptic type.

• 2. The underlying graph G has no $K_4$ or $L_3$ minor.

• 3. There is a hyperelliptic $\Gamma '$ such that $\Gamma \sim _{C1} \Gamma '$.

  If in addition $\Gamma$ is 2-connected and unweighted, then the following is equivalent to the previous.

• 4. There is a hyperelliptic $\Gamma '$ with $\Gamma \sim _{C1} \Gamma '$ and $\Gamma '$ specializes to $\Gamma$.

Proof

The implication (1) $\Rightarrow$ (2) follows from Proposition 3.8 and Proposition 3.9. Now suppose $\Gamma$ is 2-connected and has no $K_4$ or $L_3$ minor. By Lemma 4.2, G has a HTED. By Lemma 4.3, there is a $G'$ that has a HED such that $G\sim _{C1}G'$ and $G'$ specializes to G. Moreover, $G'$ is strongly of hyperelliptic type by Lemma 4.4. By Proposition 2.2(2) and Proposition 3.4, $\ell '$ may be chosen so that $\Gamma \sim _{C1} \Gamma '$. This shows (2) $\Rightarrow$ (4), and (2) $\Rightarrow$ (3) now follows from Lemma 3.6. Finally, (3) $\Rightarrow$ (1) and (4) $\Rightarrow$ (1) follow from Lemma 3.2.$\square$