Partitionability is NP-complete

Introduction

A simplicial complex $\Delta$ is partitionable if we may partition its set of faces into parts in such a way that each part consists of the faces between some facet $\sigma$ and a face $\varphi(\sigma) \subseteq\sigma$. Thus, if $[\varphi(\sigma),\sigma]=\{\tau:\varphi(\tau)\subseteq\tau\subseteq\sigma\}$, then the required condition is that

We call the map $\varphi$ a partitioning scheme of $\Delta$. We immediately see that a partitioning scheme makes it convenient to count the faces of $\Delta$. Indeed, if $\Delta$ is pure $d$-dimensional, then the generating function of the faces $f_k$ of cardinality $k$ satisfies

where $h_k$ is the number of facets $\sigma$ so that $\varphi(\sigma)$ has cardinality $k$.

Partitionability first arose in the theses of Ball and of Provan, see [1]. It is important in the theory of shellability [22]; indeed, one definition of a shelling is of a partitioning scheme with an ordering that satisfies an additional condition. There has been some recent interest in partitionable simplicial complexes [2, 3, 4, 7, 11, 19].

The decision problem Partitionability asks whether a simplicial complex $\Delta$ (given by a list of facets) is partitionable. Kleinschmidt and Onn conjectured in [14 Section 5] (see also [13]) that Partitionability is NP-complete in dimensions two and higher, and moreover that the problem remains NP-complete when restricted to pseudomanifolds. Noble showed that Partitionability is in NP even when dimension is not restricted [15]. Kaibel and Pfetsch included the Partitionability problem as the final item in their 2002 list of algorithmic problems in polytope theory [12 Problem 35].

In this paper, I will settle this problem for general simplicial complexes.

2020 Mathematics Subject Classification. 05E45, 68Q17.

It follows easily that the same holds for $d \ge2$. On the other hand, the problem admits a polynomial algorithm in dimension one [12 Problem 35].

The proof here does not follow the approach suggested in [14]. Indeed, a computer search for a “switching complex” (see Remark 4.8) in the sense of Kleinschmidt and Onn was not successful. I will instead give a reduction directly from 1-In-3-Sat. A main ingredient over the approach of Kleinschmidt and Onn will be control of the Euler characteristic of the simplicial complex produced by the reduction.

Partitionability is frequently studied in conjunction with shellable simplicial complexes. Indeed, any shellable complex is partitionable. The Shellability problem asks whether a simplicial complex (given by a list of facets) is shellable. Shellability was shown to be NP-complete by Goaoc, Paták, Patáková, Tancer, and Wagner [6] (see also [17, 18]). The simplicial complexes used for the reduction in [6] or in [18] are built by taking a 2-sphere for each variable, and wrapping clause gadgets partway around the equators. Truth assignments correspond to (homology facets in) the north/south hemispheres of the variable spheres. The complexes are Cohen-Macaulay by [4 Lemma 2.2], and one can check using the tools of § 5 below that the construction in [18] is always partitionable. In contrast, the simplicial complexes that we will build will have trivial Euler characteristic, but highly nontrivial homology; thus, they will never be Cohen-Macaulay.

The Stanley depth of a simplicial complex is a numeric parameter arising from commutative algebra considerations [21 Section 5], see [8]. It extends partitionability in the sense that a pure $d$-dimensional simplicial complex is partitionable if and only if it has Stanley depth of $d + 1$ (equivalently, Stanley depth $\ge d + 1$) [8 Section 4]. The associated Stanley Depth decision problem asks, given a simplicial complex $\Delta$ and a parameter $k$, is the Stanley depth of $\Delta$ at least $k$?

Ichim, Katthän, and Moyano-Fernández give an algorithm in [10] to find Stanley depth, but make no particular claim of efficiency.

Organization

The paper is organized as follows. In § 3, we give background material on simplicial complexes and partitionability. In § 4, we construct the necessary simplicial complex gadgets and describe the reduction. In § 5, we introduce the notion of a partitioning with gluing, used to localize and combine partitionings. In § 6, we prove Theorem 1.1. In § 7, we make additional remarks and discuss open questions.

AI disclosure

A precursor of the $V^{0}$ gadget was discovered in a computer search carried out by Claude Opus 3.5, and the argument was then developed and refined in conjunction with Opus and Claude Fab­le 5.1 models. Figures were drawn by hand and converted to TikZ by Opus. The paper is human written.

Acknowledgements

Discussing partitionability with Andrés Santamaría-Galvis has been enlightening over the years. I thank the University of Michigan Department of Mathematics for its hospitality during the 2026–2027 academic year.

Preliminaries

We assume that the reader has general familiarity with NP-completeness, on the level of [16] or similar. We review basics of simplicial complexes.

A simplicial complex is simply a hereditary set system: a family of sets $\Delta$ (called faces) so that if $\sigma\in\Delta$ and $\tau\subseteq\sigma$, then also $\tau\in\Delta$. Simplicial complexes are important in discrete geometry, as we may identify a singleton set with a vertex, a doubleton with an edge, a 3-element set with a triangle, and so forth. A simplicial complex is pure $d$-dimensional if all of its facets (maximal faces) have the cardinality $d+1$, and so dimension $d$. All simplicial complexes discussed in this paper are finite.

The main setting that we focus on is a pure 2-dimensional complex, which we see consists of a set of triangles, together with all edges and vertices that are contained in a triangle. The empty set is a face of every non-trivial simplicial complex.

A relative simplicial complex is a pair $(\Delta, \Sigma)$, where $\Delta$ is a simplicial complex and $\Sigma$ is a simplicial complex with $\Sigma\subseteq\Delta$. As a set system, we consider $(\Delta, \Sigma)$ as consisting of the faces of $\Delta$ that are not in $\Sigma$. Thus, a relative simplicial complex is partitionable if the set system $(\Delta, \Sigma)$ may be partitioned as in the introduction:

(1) goes through without change.

We will occasionally need to distinguish between the simplicial complexes $\Sigma= \varnothing$ (with no faces) and $\Sigma= \{\varnothing\}$ (having the empty set as its unique face).

The reduced Euler characteristic of the simplicial complex $\Delta$ is given by

where $f_i$ is the number of faces in $\Delta$ of cardinality $i$. Thus, the reduced Euler characteristic includes the empty set $f_0$ in the alternating sum of face numbers. Since we have $f_0 = 1$ except in the degenerate case, we see that $\widetilde{\chi}(\Delta)$ is one smaller than the (unreduced) Euler characteristic $\chi(\Delta)$. The (reduced) Euler characteristic of a relative simplicial complex $(\Delta, \Sigma)$ has the same definition, where $f_i$ is the number of faces of $\Delta$ that are not faces of $\Sigma$. Thus, $\chi(\Delta, \varnothing) = \widetilde{\chi}(\Delta)$ while $\chi(\Delta, \Sigma) = \widetilde{\chi}(\Delta) - \widetilde{\chi}(\Sigma) = \chi(\Delta) - \chi(\Sigma)$ when $\Sigma\ne\varnothing$.

We record that the $h_{d+1}$ entry in equation (1.1) has a nice description:

Proof. Plug in $x=-1$.

In particular, if $\chi(\Delta,\Sigma)=0$ and $\varphi$ is a partitioning of $(\Delta,\Sigma)$, then there is no facet $\sigma$ with $\varphi(\sigma)=\sigma$.

Passing from 2-dimensional complexes to higher-dimensional complexes is not difficult. The cone of the simplicial complex $\Delta$ is the simplicial complex $\operatorname{Cone}\Delta$ which adds one new apex vertex $a$ to the vertex set of $\Delta$, and whose facets are $\{\sigma\cup a:\sigma\text{ facet of }\Delta\}$. I did not find the following lemma in the literature in this form, although [9 Lemma 3.6] generalizes to Stanley depth of modules.

Proof. Let $a$ be the apex vertex of $\operatorname{Cone}\Delta$. If $\varphi$ is a partitioning of $\Delta$, then the map sending $\sigma\cup a$ to $\varphi(\sigma)$ is a partitioning of $\operatorname{Cone}\Delta$. Conversely, if $\varphi$ is a partitioning of $\operatorname{Cone}\Delta$, then the face $\sigma$ of $\operatorname{Cone}\Delta$ lies only in the facet $\sigma\cup\{a\}$, hence $\varphi(\sigma\cup a)\subseteq\sigma$. In particular, the map sending $\sigma$ to $\varphi(\sigma\cup a)$ is a well-defined map from facets of $\Delta$ to faces of $\Delta$, and is easily seen to be a partitioning of $\Delta$. $\square$

In particular, Corollary 1.2 follows immediately from Theorem 1.1.

The reduction

We provide a polynomial time reduction from the 1-In-3-Sat problem [20] (see also [5 LO4]). This problem takes as input a Boolean formula in the form

where each literal $\ell_i$ is either a Boolean variable $x$ or its negation $\neg x$, and asks whether there is an assignment of truth values so that each clause $(\ell_{3k+1}\lor\ell_{3k+2}\lor\ell_{3k+3})$ has exactly one true literal (equivalently, exactly two false literals). It is convenient to assume that each variable $x$ appears in at least two literals (including in negated form). This assumption causes no loss of generality, as if $x$ occurs only once, we simply repeat the clause in which it appears. See Remark 5.7 below.

Gadgets

As is usual in NP-completeness reductions from 3-Sat and its variants, the proof proceeds by building gadgets to represent variables and clauses in the simplicial complex setting. We will start by showing that Partitionability is NP-complete when expanded to relative simplicial complexes, and then show that we may restrict back to non-relative complexes.

We have three main gadgets: a variable gadget $V(k)$, a positive clause gadget $C^+$, and a negated clause gadget $C^-$, as we will now describe.

The variable gadget $V^0=V(2)$ is formed in two stages. We start by subdividing a square into four triangles $12r$, $23r$, $34r$, $14r$. We then attach the boundary of a tetrahedron to the edge $23$, and another to the edge $14$. We treat the vertex $r$ as being relative. Thus, $V^0$ has 9 vertices (1 of which is relative), 18 edges, and 12 triangles.

The variable gadget $V(k)$ is obtained from $k-1$ copies of $V^0$ by gluing the vertex $4$ from the $i$th copy to the vertex $1$ of the $(i+1)$st copy. See Figure 3.1. The positive variable vertices in $V(k)$ consist of the vertex $3$ from all copies of $V^0$ and the (unglued) vertex $1$ from the first copy of $V^0$. The negated variable vertices in $V(k)$ consist of the vertex $2$ from all copies of $V^0$ and the (unglued) vertex $4$ from the last copy of $V^0$. Thus, $V(k)$ has $k$ positive variable vertices and $k$ negated variable vertices. The relativizable vertices consist of the vertex $r$ from all copies of $V^0$.

The positive clause gadget $C^+$ is pictured in Figure 3.2. It consists of three subdivided triangles, each glued to an edge of a center triangle; see also § 6.2. The hollow vertices will later be glued to positive or negated variable vertices. We treat the boundary of $C^+$ (shown in red, and generated by the edges that are only in a single facet) as being relative.

The negated clause gadget $C^-$ is formed by gluing three copies of the complex pictured in Figure 3.2 (right) along the bolded edge. The hollow vertices will later be glued to positive or negated variable vertices. We treat the boundary of $C^-$ (shown in red, generated by edges that are in a single facet) as being relative.

Putting the gadgets together

For ease, we first describe the relative case of the simplicial complex output by the reduction. We are given a 1-In-3-SAT instance with $m$ clauses and $n$ variables. For the variable $x$, let $m_x$ be the number of occurrences of $x$ or $\neg x$ as a literal in the instance. We have assumed without loss of generality that each $m_x$ is at least 2.

For each $x$, we take $V_x$ to be a copy of $V(m_x)$. The subcomplex $R_x$ consists of all relativizable vertices (together with the empty set).

For each clause $\ell_1 \vee\ell_2 \vee\ell_3$, we take both a copy of $C^+$ and also a copy of $C^-$. We notice that exactly one of $\ell_1,\ell_2,\ell_3$ is true if and only if exactly two of $\neg\ell_1,\neg\ell_2,\neg\ell_3$ are true. We glue the three hollow vertices of $C^+$ to variable vertices corresponding to the literals $\ell_1,\ell_2,\ell_3$, and glue the three hollow vertices of $C^-$ to the variable vertices corresponding to the negated literals $\neg\ell_1,\neg\ell_2,\neg\ell_3$.

Thus, for each $x$ appearing in $\ell_1,\ell_2,\ell_3$, we have glued to both a positive variable vertex and also to a negated variable vertex in $V_x$. The gluing yields a bijection between variable vertices (taken over all $V_x$) and hollow vertices (taken over all positive and negated clause gadgets).

Our relative complex thus consists of

It is easy to extend this to non-relative simplicial complexes: given $\Delta$ and $\Sigma$ as above, we let $\Delta^{+}$ be obtained from $\Delta$ by adding a disconnected triangle, gluing an independent copy of the triangulated 5-vertex Möbius strip to each vertex of $\Sigma$, and gluing an independent copy of the tetrahedral boundary to each edge of $\Sigma$.

It is clear that the reduction may be implemented in polynomial time. Theorem 1.1 is an immediate consequence of Corollary 3.2.

Size of the reduction

We count the triangles in $\Delta$ and in $\Delta^{+}$. We note that $C^{+}$ has 9 vertices (6 of them relative), 18 edges (6 relative), and 10 triangles; $C^{-}$ has 11 vertices (8 relative), 22 edges (9 relative), and 12 triangles.

Each $V_x$ contributes $(m_x-1)\cdot12$ triangles to $\Delta$, while each clause contributes $10+12$ triangles in $C^{+}$ and $C^{-}$ to $\Delta$. Noting that $\sum m_x=3m$, the relative complex $(\Delta,\Sigma)$ has $58m-12n$ triangles.

Finally, each vertex of $\Sigma$ contributes 5 additional triangles to $\Delta^{+}$ (using the 5 triangle Möbius strip), each edge contributes 4 additional triangles, and the empty set contributes 1 additional triangle. Thus, each variable $x$ contributes $(m_x - 1) \cdot5$ additional triangles, and each clause contributes $14 \cdot5 + 15 \cdot4$ triangles. We see that the simplicial complex $\Delta^{+}$ has $1 + 203m - 17n$ triangles.

Partitioning complexes with gluing

The complexes $(\Delta, \Sigma)$ and $\Delta^{+}$ of Theorem 3.1 and Corollary 3.2 are built by gluing together gadgets along uncomplicated subcomplexes. In this section, we develop language to describe each “piece” of such a partitioning scheme. The approach here will offer better control of gluing along small intersections as compared to e.g. [4 Theorem 3.1].

Let $(K, \Sigma)$ be a relative simplicial complex, and let $\Gamma$ be another subcomplex of $K$. It is convenient to require that no face of $\Gamma$ be a facet of $K$. We wish to think of $K$ as sitting inside a larger (relative) simplicial complex as a full-dimensional subcomplex, and $\Gamma$ as where $K$ is glued to the rest of the complex.

We say that a partitioning with gluing at $\Gamma$ of $(K, \Sigma)$ is a map $\varphi$ from facets of $(K, \Sigma)$ to faces of $(K, \Sigma)$ where $\varphi(\sigma) \subseteq\sigma$ and so that

1. The sets of faces $[\varphi(\sigma), \sigma]$ are disjoint over facets $\sigma$ of $(K, \Sigma\cup\Gamma)$.

2. Each face of $(K, \Sigma\cup\Gamma)$ is in exactly one of the $[\varphi(\sigma), \sigma]$.

Thus, each face of $(\Gamma, \Gamma\cap\Sigma)$ is in at most one of the $[\varphi(\sigma), \sigma]$. If a face $\gamma$ of $\Gamma$ is in one of the $[\varphi(\sigma), \sigma]$, then we say it is used; otherwise it is unused.

Thus, if a $d$-dimensional complex may be written as $K \cup\Lambda$, where $K$ and $\Lambda$ are both $d$-dimensional and $K \cap\Lambda$ is lower dimensional, then a partitioning of $K \cup\Lambda$ induces a partitioning with gluing at $K \cap\Lambda$ of $K$, and also a partitioning with gluing at $K \cap\Lambda$ of $\Lambda$. Given partitionings with gluing of $K$ and of $\Lambda$, we may be able to recover a partitioning of $K \cup\Lambda$ if every face of the intersection is used in one or the other of $K$ or $\Lambda$.

Indeed, the following follows immediately from definition:

Remark 4.2. Conversely, in the situation of Proposition 4.1, a partitioning of $(K, \Sigma)$ restricts to partitionings with gluing of the constituent $(K_i, K_i \cap\Sigma)$’s with the “used exactly once” property.

We are interested in putting restrictions on the used faces of a partitioning with gluing of an uncomplicated simplicial complex. We give several examples, which we will make good use of later.

Remark 4.6. The boundary of a tetrahedron is easily seen to admit both types of partitioning with gluing at an edge from Lemma 4.5.

As a consequence of these examples, we obtain:

Remark 4.8. The switching complex desired by Kleinschmidt and Onn [14, 13] may be formulated in this language: it is a simplicial complex $K$, with gluing subcomplex $\Gamma= \{\varnothing, v_1, v_2, v_3\}$ such that any partitioning with gluing at $\Gamma$ of $(K, \{\varnothing\})$ uses either all or none of $v_1, v_2, v_3$, and so that both types of partitionings with gluing exist.

The variable gadgets of this paper have a similar rigidity with respect to partitionings with gluing, so long as they are embedded in a simplicial complex with trivial Euler characteristic.

Proof of the main theorems

Throughout this section, let $(\Delta, \Sigma)$ be as in Theorem 3.1, and $\Delta^+$ be as in Corollary 3.2. In § 6.1, we will show that the relative Euler characteristic $\chi(\Delta, \Sigma)$ is zero. In § 6.2, we will show that this highly constrains partitionings with gluing of our gadgets: in particular, an appropriate partitioning with gluing of the variable gadget $(V_x, R_x)$ will use all positive or all negated variable vertices, corresponding to the two truth values of $x$. In § 6.3 we will assemble these ingredients into a proof of Theorem 3.1.

Relative Euler characteristic

We begin by computing the relative Euler characteristic of $(\Delta, \Sigma)$, which will be useful in light of Lemma 2.1.

Since $V^0$ is obtained by gluing together two spheres and a disc along contractible intersections, we have $\widetilde{\chi}(V^0) = 2$. Since $C^+$ and $C^-$ are contractible, $\partial C^+$ is an $S^1$, and $\partial C^-$ is a bouquet of two $S^1$’s, we have $\chi(C^+, \partial C^+) = 1$ and $\chi(C^-, \partial C^-) = 2$.

As $V_x$ is homotopy equivalent to a bouquet of $m_x - 1$ copies of $V^0$, each variable gadget $(V_x, R_x)$ contributes $(2m_x - 2) - (m_x - 2) = m_x$ to the relative Euler characteristic. Gluing $(C^+, \partial C^+)$ along three vertices contributes $1 - 3 = -2$ to the relative Euler characteristic, and gluing $(C^-, \partial C^-)$ along three vertices contributes $2 - 3 = -1$. Here, we have already counted the hollow vertices of $C^+$ and $C^-$ in the $V_x$ gadgets. Our Euler characteristic of the relative complex is

Partitioning the gadgets

The essential ingredient in the proof is the $V^0$ relative simplicial complex.

Proof. By Lemma 4.5, if there is no facet $\sigma$ with $\varphi(\sigma)=\sigma$, then the restrictions of $\varphi$ to the two tetrahedral boundaries use the edges 23 and 14. By counting edges, $\varphi$ must send two facets to vertices and two to edges. The result now follows by inspection. See Figure 5.1. □

Since $V_x$ is obtained by gluing together many copies of $V^{0}$, we obtain immediately the following.

Proof. If (a) does not occur, then by Lemma 5.1 the vertices shared by adjacent copies of $V^{0}$ are either all used by the predecessor copy of $V^{0}$ or all by the successor copy. □

We now consider the clause gadgets. Let $\mathrm{T}$ be the simplicial complex with vertices 1, 2, 3, $\ell$ and triangles $12\ell$, $23\ell$, $13\ell$, and $\Sigma_{\mathrm{T}}$ the subcomplex generated by 12 and 13. Thus, the relative complexes $(C^{+},\partial C^{+})$ respectively $(C^{-},\partial C^{-})$ are formed by gluing together three copies of $(\mathrm{T},\Sigma_{\mathrm{T}})$ around a triangle respectively around three triangles sharing a common edge.

Proof. If there is no facet with $\varphi(\sigma)=\sigma$, then count edges! If $\ell$ is not used, then the three edges $1\ell$, $2\ell$, $3\ell$ are covered by three facets. If $\ell$ is used, then $[\ell,\sigma]$ covers two edges, and we need two additional edges to cover the remaining two facets. □

Proof. Assume that there is no facet with $\varphi(\sigma)=\sigma$. The partitioning $\varphi$ matches the middle triangle to one of its edges (as its vertices are relative). By three applications of Lemma 5.3, $\varphi$ does not use $\ell_i$ in the corresponding copy of $\mathrm{T}$, but does use $\ell_i$ in the other two copies. □

The corresponding result for $C^{-}$ is proved in exactly the same way.

All of the possibilities for partitionings with gluing enumerated in the lemmas and corollary of this section can indeed be constructed.

Proof. Inspection. □

Proof of Theorem 3.1

We are now ready to prove Theorem 3.1. First, suppose that $\varphi$ is a partitioning of $(\Delta, \Sigma)$. Then $\varphi$ induces a partitioning with gluing of each variable and clause gadget:

• In each variable gadget $(V_x, R_x)$, we have an induced partitioning with gluing at the subcomplex generated by the variable vertices.

• In each positive clause gadget $(C^{+}, \partial C^{+})$ and negated clause gadget $(C^{-}, \partial C^{-})$, we have an induced partitioning with gluing at the subcomplex generated by the three hollow vertices.

Since the Euler characteristic $\chi(\Delta, \Sigma)$ is zero, Lemma 2.1 tells us that no facet $\sigma$ has $\varphi(\sigma) = \sigma$. It follows from Corollary 5.2 that each $(V_x, R_x)$ uses either positive or negated variable vertices. Setting $x$ true if $(V_x, R_x)$ uses the positive variable vertices (and otherwise false) yields a truth assignment $\mathcal{T}$.

The remaining variable vertices must be used by the clause gadgets (both positive and negated) corresponding to clauses containing $x$ or $\neg x$ as a literal. By Lemmas 5.4 and 5.5, each $(C^{+}, \partial C^{+})$ must use exactly two of these variable vertices left over from $\mathcal{T}$ and each $(C^{-}, \partial C^{-})$ must use exactly one. Since $C^{+}$ attaches to the literals of the associated clause and $C^{-}$ to the literals of the negated clause, we see that $\mathcal{T}$ is a legal 1-In-3-Sat truth assignment.

Conversely, if $\mathcal{T}$ is a 1-In-3-Sat truth assignment, then we put together partitionings with gluing of each gadget using Proposition 4.1. Since $\chi(\Delta, \Sigma) = 0$, we will not want any facets with $\varphi(\sigma) = \sigma$ by Lemma 2.1.

We take a partitioning with gluing of each $(V_x, R_x)$ that uses $x$ or $\neg x$ according to whether $x$ is true or false in $\mathcal{T}$. Since $\mathcal{T}$ has exactly one true literal in each clause, we can use the remaining variable vertices in $(C^{+}, \partial C^{+})$ and $(C^{-}, \partial C^{-})$ gadgets.

Corollary 3.2 is an immediate consequence of Theorem 3.1 and Proposition 4.7.

Remark 5.7. The reduction assumes that a 1-In-3-Sat instance has every variable occurring in at least two literals of clauses. It is possible to avoid this assumption, using a single vertex relative to the empty set as a $V(1)$ gadget. As this approach would require more care with the partition gluing arguments, we do not follow it further.

Additional remarks and questions

Partitionability has often been studied in the context of Cohen–Macaulay simplicial complexes [22]. Indeed, it was a long-standing conjecture disproved by Duval, Goeckner, Klivans and Martin [4] that every Cohen–Macaulay simplicial complex is partitionable. I do not know if partitionability is NP-complete when restricted to Cohen–Macaulay complexes.

Examples of Cohen–Macaulay simplicial complexes that are not partitionable seem to be relatively difficult to construct. In [4], the authors construct a 3-dimensional complex with 42 tetrahedra. No examples seem to be known of 2-dimensional complexes that are Cohen–Macaulay but not partitionable, and for example [19] shows that every triangulation of the dunce cap or of $\mathbb{RP}^{2}$ is partitionable.

Of course, a “yes” answer to Question 6.2 would mean that Question 6.1 is only interesting in dimensions 3 and higher.

The full conjecture of Kleinschmidt and Onn [14], that Partitionability is NP-complete already for 2-dimensional pseudomanifolds, also remains open. In a 2-dimensional pseudomanifold, every edge lies in at most two triangles. Our construction here has edges that lie in exactly three triangles in two places: the edges where we attach tetrahedral boundaries, and an edge in the $C^{-}$ gadget.

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