On approximation of asymmetric separators of the n-cube

Abstract

A new combinatorial result intertwined with the Brouwer fixed point theorem for the n-cube is given. This result can be used for any map (f₁, ..., fn): [0, 1]ⁿ → [0, 1]ⁿ to approximate the components of the set {(x₁, . . . , x _n ) ∈ [0, 1]ⁿ : f _i (x₁, . . . , x _n ) = x _i } that separate the n-cube between the i th opposite faces. Equivalently, for maps g : [0, 1]ⁿ → ℝ such that g(x)g(y) ≤ 0 for any x ∈ {0} × [0, 1]^n-1and y ∈ {1} × [0, 1]^n-1, one can use the algorithm to approximate the components of g^-1(0) that separate [0, 1]ⁿ between {0} × [0, 1]^n-1and {1} × [0, 1]^n-1. The methods are based on an earlier result of P. Minc and the present authors and relate to results of several other authors such as Jayawant and Wong, Kulpa and Turzański, and Gale.

Mathematics Subject Classification (2000): Primary 54H25; 54-04; Secondary 55M20; 54F55.

1. Introduction

In [1] Minc and the present authors described combinatorial methods that allow approximation of connected symmetric separators of the n-sphere and n-cube. The symmetric separators arise in the context of the Borsuk-Ulam antipodal theorem and a theorem of Dyson for the 2-sphere [2, 3]. The purpose of the present paper is to show how the results from [1] can be extended to the setting of asymmetric separators and the Brouwer fixed point theorem for the n-cube. The classic result of L.E.J. Brouwer says that the n-dimensional cube Iⁿ = [0, 1]ⁿ has the fixed point property; that is, for any map^a f : Iⁿ → Iⁿ, there is x ∈ Iⁿ such that f(x) = x. There are many important applications of the Brouwer's theorem such as, for example, those concerning existence of solutions for differential equations [4], or equilibrium strategies in multi-person games relating to market problems in economics [5]. This is why computability of fixed points became an important theme in the fixed point theory. The first fixed point algorithm was given by Scarf [6]. Soon after, other were given by Eaves [7] and Todd [8] (see, for example, [9, 10] for a comprehensive treatment of this subject with applications). There are also several combinatorial equivalents of Brouwer's theorem. The best known is probably Sperner's lemma [11] on coloring vertices of a barycentric subdivision of an n-simplex. Some other transfer the fixed point problem to the scenario of board games, such as Hex [12] or Chess [13].

In the present paper, in Theorem 3.1, we formulate yet another combinatorial result that implies the Brouwer fixed point theorem. Its baby version can be formulated as follows.

Theorem 1.1. Suppose f : V→ ℝ is a function defined on the set of vertices V of a triangulation T of Iⁿ. Suppose in addition that f(v₁)f(v₂) ≤ 0 for any vertices v₁ ∈ {0} × I^n-1and v₂ ∈ {1} × I^n-1. Then, there is a subcollection S ⊆ T of simplices of dimension n such that

• 1. for every simplex σ ∈ S there is an edge [v, u] such that f (v) f (u) ≤ 0;

• 2. ∪S separates Iⁿ between {0} × I^n-1and {1} × I^n-1.

The above theorem implies the Brouwer fixed point theorem in the following way. If (f₁, . . . , f _n ): Iⁿ→ Iⁿis a map and $\mathcal{X}$ is a polyhedral complex with Iⁿ as its underlying space, then each g _i (x₁, . . . , x _n ) = f _i (x₁, . . . , x _n ) -x _i satisfies the assumptions of Theorem 1.1 and one can find C _i , an approximation of a component of $g_i^{-1}\left(0\right)$, that separates Iⁿ between the i th opposite (n - 1)-dimensional faces. By Eilenberg-Otto theorem (see [14]) ${\bigcap }_{i=1}^n{C}_i$ is nonempty and approximates a fixed point of f.

We give a stronger (but at the same time more technical) version of the above result in Theorem 3.1, and in Section 4, we show how along with Theorem 4.1 it can be used to approximate a connected separating component of the set of zeros of an arbitrary map f : Iⁿ → ℝ, which assumes opposite signs on some two opposite (n - 1)-faces of Iⁿ. The case when n = 2 was already considered in [15]. The methods used in the proof of Theorem 3.1 are based on those introduced in [1] where, in connection with the Borsuk-Ulam antipodal theorem, it was shown how to approximate a connected separator of the n-sphere Sⁿ(or Iⁿ), invariant under the antipodal map. Any such separator was corresponding to a component of f^-1(0), with f : Sⁿ → ℝ (or f : Iⁿ→ ℝ) an odd map (related combinatorial results can be found in [2, 16]). However, the methods of [1] were dealing only with symmetric separators and are insufficient in the case of arbitrary separators. First, unlike in the case of symmetric separators and odd maps, if a map f : Iⁿ→ R satisfies the condition f({1} × I^n-1) ⊆ [0, ∞) and f({0} × I^n-1) ⊆ (-∞, 0] for some i = 1, . . . , n, there may be no unique connected separator of Iⁿin f^-1(0). Clearly, f^-1(0) may consist of several disjoint separating components, none of which needs to be symmetric. Second, the algorithms in [1] were making use of the fact that the symmetric component of f^-1(0) is the separating omponent, when f is odd. Therefore, if a subcollection of the triangulation approximated a component of f^-1(0) and, at the same time, was symmetric, this was sufficient to determine that it separated Sⁿ(or Iⁿ). This is why one is forced to develop new combinatorial criteria for arbitrary separators in Iⁿ. In Section 4 we furnish such a computer implementable criterion that allows isolating those subcollections of the triangulation, approximating a component of f^-1(0), that separate Iⁿ from those that do not.

2. Preliminaries

For a collection of sets $\mathcal{K}$, by ${\mathcal{K}}^{*}$ we will denote the union of all its elements. π _i : [0, 1]ⁿ → [0, 1] will denote the projection onto the i th coordinate. $I_i^{+}$ and $I_i^{-}$ will denote the i th opposite (n - 1)-dimensional faces of Iⁿ, that is $I_i^{+}={\pi }_i^{-1}\left(1\right)$and $I_i^{-}={\pi }_i^{-1}\left(0\right)$. C separates Iⁿ (or is a separator of Iⁿ) between $I_i^{+}$ and $I_i^{-}$ if for any $x\in I_i^{+}\backslash C$,$y\in I_i^{-}\backslash C$, there are U, V , distinct components of Iⁿ\C, such that x ∈ U and y ∈ V. A map $g:{\mathcal{X}}^{*}\to ℝ$ is piecewise linear if given {σ _j : j = 1, . . . , N}, a triangulation of ${\mathcal{X}}^{*}$, for every j the restriction of g to the simplex σ _j is linear, that is $g\left({\sum }_{i=1}^k{\lambda }_i{a}_i\right)={\sum }_{i=1}^k{\lambda }_{i}g\left(a_i\right)$ where a₁, . . . , a _k are the vertices spanning σ _j and λ _i ≥ 0 with ${\sum }_{i=1}^k{\lambda }_i=1$ (see [17]).

We will heavily rely on the following inductive procedure introduced by Minc and the two authors in [1]. Let $\mathcal{X}$ be a polyhedral complex such that ${\mathcal{X}}^{*}={\left[0,1\right]}^n$. Let $\mathcal{V}\left(\mathcal{X}\right)$ and $\mathcal{E}\left(\mathcal{X}\right)$ denote the collections of vertices and edges, respectively. Suppose $f:\mathcal{V}\left(\mathcal{X}\right)\to ℝ$ is a function. Let ${\mathcal{E}}_f$ be the collection of those edges $e=⟨u,v⟩\in \mathcal{E}\left(\mathcal{X}\right)$ that f (u) f (v) ≤ 0. Let $\mathcal{P}\subset \mathcal{X}$ be the collection of polytopes in $\mathcal{X}$ of dimension n. For any $e\in {\mathcal{E}}_f,$ $\mathcal{C}\left(e\right)$ is defined by induction.

• Let ${\mathcal{C}}_0\left(e\right)$ be the collection of those $P\in \mathcal{P}$ that contain e.

• Suppose ${\mathcal{C}}_{i-1}\left(e\right)$ has been defined. Define ${\mathcal{C}}_i\left(e\right)$ to be the collection of those $P\in \mathcal{P}$ such that the intersection $P\cap {\mathcal{C}}_{i-1}{\left(e\right)}^{*}$ contains an edge from ${\mathcal{E}}_f$ or a vertex from f^-1 (0).

Clearly, ${\mathcal{C}}_{i-1}\left(e\right)\subset {\mathcal{C}}_i\left(e\right)$ and there is an integer q ≥ 0 such that ${\mathcal{C}}_q\left(e\right)={\mathcal{C}}_{q+1}\left(e\right).$ For the first such number q (e) set $\mathcal{C}\left(e\right)={\mathcal{C}}_{q\left(e\right)}\left(e\right)$. Note that $e\subset \mathcal{C}{\left(e\right)}^{*}$, and $\mathcal{C}{\left(e\right)}^{*}$ is connected.

3. Combinatorial theorem on separators of Iⁿbetween opposite faces

Let χ be a polyhedral complex such that ${\mathcal{X}}^{*}={\left[0,1\right]}^n$. Note that $\mathcal{X}$ can be subdivided to give a triangulation of Iⁿ, without introducing new vertices [18], and consequently, every function $f:\mathcal{V}\left(\mathcal{X}\right)\to ℝ$ has a piecewise linear extension g : Iⁿ → ℝ. The following result is of purely combinatorial nature.

Theorem 3.1. Suppose that $f:\mathcal{V}\left(\mathcal{X}\right)\to ℝ$ is a function satisfying

$$f\left(\mathcal{V}\left(\mathcal{X}\right)\cap I_i^{+}\right)\subseteq \left[0,+\infty \right),f\left(\mathcal{V}\left(\mathcal{X}\right)\cap I_i^{-}\right)\subseteq \left(-\infty ,0\right]$$

(3.1)

for some i ∈{1, . . . ,n}. Let $\mathcal{L}$ be a subcollection of $\mathcal{E}\left(\mathcal{X}\right)$ such that ${\mathcal{L}}^{*}\subseteq I_j^{\epsilon }$ for some j ≠ i andε ∈ {+, -}, and ${\mathcal{L}}^{*}$ is an arc with endpoints in $I_i^{+}$ and $I_i^{-}$. Then, there is an edge $d\in \mathcal{E}\left(\mathcal{X}\right)\cap \mathcal{L}$ such that $d\in {\mathcal{E}}_f$ and

• 1. each $C\in \mathcal{C}\left(d\right)$ contains an edge from ${\mathcal{E}}_f$,

• 2. $\mathcal{C}{\left(d\right)}^{*}$ separates Iⁿ between $I_i^{+}$ and $I_i^{-}$ and

• 3. for any other such $d^{\prime }\in \mathcal{L}$ with $\mathcal{C}{\left(d^{\prime }\right)}^{*}$ satisfying ( 1)-( 2), either $\mathcal{C}\left(d^{\prime }\right)\cap \mathcal{C}\left(d\right)=\varnothing$ or $\mathcal{C}\mathsf{\text{(}}{d}^{\prime }\mathsf{\text{) = }}\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$.

Proof. Without loss we can assume that i = 1. Let g : Iⁿ → ℝ be a piecewise linear extension of f. Then, g is continuous and g(v) = f(v) for any $v\in \mathcal{V}\left(\mathcal{X}\right)$.

Claim 3.1.1. If $K\in \mathcal{X}$, then g(r) = 0 for some r ∈ K if and only if there is an edge $\left[w,v\right]\in \mathcal{E}\left(\mathcal{X}\right)$ such that $w,v\in K\cap \mathcal{V}\left(\mathcal{X}\right)$ and f (w) f (v) ≤ 0.

Proof of claim 3.1.1. First suppose $K\in \mathcal{X}$ is such that there are vertices $w,v\in K\cap \mathcal{V}\left(\mathcal{X}\right)$, and f (w) f (v) ≤ 0. Then, either f(w)f(v) = 0 or f(w)f(v) < 0. In the first case, clearly g(w) = f(w) = 0 or g(v) = f(v) = 0. Otherwise there must be r ∈ [u, v] such that g(r) = 0. For the converse, suppose g(r) = 0 for some r ∈ K. Then, $r={\sum }_{i=1}^k{\lambda }_i{a}_i$, where a _o , . . . , a _k are vertices of K spanning a simplex σ ⊆ K, and λ _i ≥ 0 with ${\sum }_{i=1}^k{\lambda }_i=1$. Therefore, $0=g\left(r\right)={\sum }_{i=1}^k{\lambda }_{i}g\left(a_i\right)={\sum }_{i=1}^k{\lambda }_{i}f\left(a_i\right)$. Clearly, there is l such that f(a _l ) = 0, or there are a _j , a _t such that f(a _j )f(a _t ) < 0.   □

Claim 3.1.2. $g\left(I_i^{+}\right)\subseteq \left[0,+\infty \right),g\left(I_i^{-}\right)\subseteq \left(-\infty ,0\right]$.

Proof of claim 3.1.2. Similarly to the proof of claim 3.1.1, this follows from the fact that if a _o , . . . , a _k spans a simplex σ and g (a _i ) ≥ 0 (g(a _i ) ≤ 0) for every i, then ${\sum }_{i=1}^k{\lambda }_{i}g\left(a_i\right)\ge 0\left({\sum }_{i=1}^k{\lambda }_{i}g\left(a_i\right)\le 0\right)$ for λ _i ≥ 0. Consequently, g(σ) ⊆ [0,+∞) (g(σ) ⊆ (-∞, 0]) for any such σ.   □

Now, consider the following decomposition of the n-cube.

Z = {x ∈ Iⁿ : g(x) = 0}, N = {x ∈ Iⁿ : g(x) < 0}, P = {x ∈ Iⁿ : g(x) > 0}. Clearly Z separates Iⁿ between P and N. Let Z₁, . . . , Z _p be the components of Z. It is well known that if X is a connected, locally connected and unicoherent space then any closed set separating X contains a connected subset separating X ([[19], p. 195], cf. [[20], p. 429, Theorem 1.(vi)]). Since Z is closed and separates Iⁿ, by unicoherence of Iⁿ, there must be q such that Z _q separates Iⁿ between N and P. Consequently, Z _q separates Iⁿ between $I_i^{+}$ and $I_i^{-}$ by claim 3.1.2.

Consider $\mathcal{S}$ a subcollection of $\mathcal{K}$ such that $K\in \mathcal{S}$ if, it has nonempty intersection with Z _q , that is $\mathcal{S}=\left\{K\in \mathcal{X}:K\cap Z_q\ne \varnothing \right\}$. Clearly $Z_q\subseteq {\mathcal{S}}^{*}$ and therefore ${\mathcal{S}}^{*}$ separates Iⁿ between $I_i^{+}$ and $I_i^{-}$.

Now, let $\mathcal{L}$ be a subcollection of $\mathcal{E}\left(\mathcal{X}\right)$ such that ${\mathcal{L}}^{*}\subseteq I_j^{\epsilon }$ for some j ≠ 1 and ε ∈ {+, -}, and ${\mathcal{L}}^{*}$ is an arc with endpoints $a\in I_1^{+}$ and $b\in I_1^{-}$, that is ${\mathcal{L}}^{*}=\left[a,b\right]$. Since Z _q ∩ Bd (Iⁿ) separates Bd (Iⁿ) between $I_i^{+}$ and $I_i^{-}$, we conclude there is z ∈ [a, b] ∩ Z _q . Additionally, there is $d\in \mathcal{L}$ such that z ∈ d. By claim 3.1.1 $d\in {\mathcal{E}}_f\cap \mathcal{L}$, and since d ∩ Z _q ≠ ∅ therefore $d\in \mathcal{S}$.

Claim 3.1.3. $S\subseteq \mathcal{C}\left(d\right)$.

Proof of claim 3.1.3. Let $L\in \mathcal{S}$ be such that d ∈ L. Clearly $L\in {\mathcal{C}}_0\left(d\right)\subseteq \mathcal{C}\left(d\right)$. Heading toward a contradiction suppose $\mathcal{S}\backslash \mathcal{C}\left(d\right)\ne \varnothing$. Consider a partition of $\mathcal{S}$ into the following two sets

$${\mathcal{S}}_1=\left\{T\in \mathcal{S}:T\in \mathcal{C}\left(d\right)\right\},{\mathcal{S}}_2=\left\{T\in \mathcal{S}:T\notin \mathcal{C}\left(d\right)\right\}.$$

By definition of $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$, for any $T\in {\mathcal{S}}_2$ and for any $\tilde{T}\in {\mathcal{S}}_1$, we must have that whenever $s\in \mathcal{E}\left(\mathcal{X}\right)$ and $s\subseteq T\cap \tilde{T}$ then $s\notin {\mathcal{E}}_f$. Otherwise T would be in ${\mathcal{S}}_1$. Therefore, $\left(T\cap \tilde{T}\right)\cap Z_q=\varnothing$, by claim 3.1.1. Consequently, there is a partition of Z _q into two disjoint sets $Z_q\cap {\mathcal{S}}_1^{*}$ and $Z_q\cap {\mathcal{S}}_2^{*}$. Since both are closed, we obtain a contradiction with connectedness of Z _q .   □

Now, property (1) is an immediate consequence of the definition of $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$. Since ${\mathcal{S}}^{*}{\subseteq }^{*}\mathcal{C}\left(d\right)$, (2) easily follows from the fact that ${\mathcal{S}}^{*}$ separates Iⁿ between $I_i^{+}$ and $I_i^{-}$. Now, suppose $d^{\prime }\in \mathcal{L}$ is another edge with $\mathcal{C}{\left(d^{\prime }\right)}^{*}$ satisfying (1)-(2). If $\mathcal{C}\left(d^{\prime }\right)\cap \mathcal{C}\left(d\right)\ne \varnothing$, then there is K such that $K\in {\mathcal{C}}_j\left(d\right)$ and $K\in {\mathcal{C}}_p\left(d^{\prime }\right)$ for some j and p. Consequently, ${\bigcup }_{i=0}^j{\mathcal{C}}_i\left(d\right)\subseteq {\bigcup }_{i=p}^{q\left(d^{\prime }\right)}{\mathcal{C}}_i\left(d^{\prime }\right)$ and ${\bigcup }_{i=j}^{q\left(d\right)}{\mathcal{C}}_i\left(d\right)\subseteq {\bigcup }_{i=p}^{q\left(d^{\prime }\right)}{\mathcal{C}}_i\left(d^{\prime }\right)$, by definition of $\mathcal{C}\left(d^{\prime }\right)$. Clearly $\mathcal{C}\left(d\right)\subseteq \mathcal{C}\left(d^{\prime }\right)$. Similarly $\mathcal{C}\left(d^{\prime }\right)\subseteq \mathcal{C}\left(d\right)$. That justifies (3) and completes the proof.   □

4. Algorithm approximating connected separators of Iⁿ

Suppose $\mathcal{K}$ is a partition of Iⁿ into kⁿ congruent n-cubes, all with side length equal to $\frac{1}{k}$. In this section we shall furnish a computer implementable criterion for the union of a subcollection of $\mathcal{K}$ to separate Iⁿ between some two opposite faces. Suppose $\mathcal{S}\subseteq \mathcal{K}$ and we want to determine if ${\mathcal{S}}^{*}$ separates Iⁿ between $I_i^{+}$ and $I_i^{-}$.

Let $K\in \mathcal{K}\backslash \mathcal{S}$ be an n-cube. G is a j-face of K if dim (G) = j and G = K ∩ L for some K, $L\in \mathcal{K}$. We will define Comp $\left(K,\mathcal{S}\right)$ by induction. Let Comp₁ $\left(K,\mathcal{S}\right)$ consists of K and all those cubes L in $\mathcal{K}\backslash \mathcal{S}$ such that K ∩ L is an (n - 1)-face. Suppose Comp _p $\left(K,\mathcal{S}\right)$ has already been defined and let Comp_p+1$\left(K,\mathcal{S}\right)$ consists of all cubes in Comp _p $\left(K,\mathcal{S}\right)$, and all those cubes R in $\mathcal{K}\backslash \mathcal{S}$ for which there is a cube L ∈ Comp _p $\left(K,\mathcal{S}\right)$ such that L ∩ R is an (n - 1)-face. Since $\mathcal{K}$ consists of only finite number of cubes Comp _q $\left(K,\mathcal{S}\right)$ = Comp_q+1$\left(K,\mathcal{S}\right)$ for some natural number q. Let q(K) be the first such number and let Comp $\left(K,\mathcal{S}\right)=\mathsf{\text{Com}}{\mathsf{\text{p}}}_{q\left(K\right)}\left(K,\mathcal{S}\right)$.

Theorem 4.1. ${\mathcal{S}}^{*}$separates Iⁿ between $I_i^{+}$ and $I_i^{-}$ iff

$$\text{Comp }{(K\text{,}\mathcal{S})}^{*}\cap I_i^{-}=\varnothing foreveryK\in \mathcal{K}suchthatK\cap I_i^{+}\ne \varnothing .$$

(4.1)

Proof. If the condition (4.1) is not satisfied, then clearly ${\mathcal{S}}^{*}$ does not separate Iⁿ between $I_i^{+}$ and $I_i^{-}$. Namely, Comp ${\left(K,\mathcal{S}\right)}^{*}$ for some K contains a connected set, disjoint with ${\mathcal{S}}^{*}$, intersecting both $I_i^{+}$ and $I_i^{-}$ in a nonempty set. For the converse, by contradiction suppose that the condition (4.1) is satisfied but ${\mathcal{S}}^{*}$ does not separate Iⁿ between $I_i^{+}$ and $I_i^{-}$. Let A be a connected component of $I^n\backslash {\mathcal{S}}^{*}$ intersecting both $I_i^{+}$ and $I_i^{-}$ in a nonempty set. Let $\mathcal{R}$ be a subcollection of $\mathcal{K}\backslash \mathcal{S}$ such that ${\mathcal{R}}^{*}$ is connected and $\mathcal{A}\subseteq {\mathcal{R}}^{*}$. Without loss of generality, we can assume that $\mathcal{R}$ is a minimal such collection. We will obtain a contradiction showing, by induction, that for any two cubes in $\mathcal{R}$ if their intersection is an m-face, then m ∉ {0, . . . , n - 2}. Suppose K, $L\in \mathcal{R}$ are two cubes such that K ∩ L ≠ ∅ but L ∉ Comp $\left(K,\mathcal{S}\right)$. Since K ∩ L must be an m-face, for some m < n, we must have that K ∩ L is an m-face with m < n - 1. Suppose m = n - 2, then there are exactly 2² - 2 other cubes sharing this m-face. Let T be any of those two cubes. Then, T ∩ K and T ∩ L are (n - 1)-faces, K ∩ L ⊆ T and T must be in $\mathcal{S}$. A contradiction with the fact that (K ∩ L) ∩ A ≠ ∅ and therefore m ≠ n - 2. Suppose we have already proved that m < n - i. We shall show that m ≠ n - (i + 1). Suppose otherwise, that is K ∩ L is an n - (i + 1)-face, for some K, $L\in \mathcal{R}$. Then, there are 2ⁱ⁺¹- 2 other cubes having this n - (i + 1)-face in common. Let T be one of them such that T ∩ K is an (n - 1)-face. Then, T ∩ L is an (n - i)-face and K ∩ L ⊆ T. Since (K ∩ L) ⊆ T and (K ∩ L) ∩ A ≠ ∅, therefore A ∩ T ≠ ∅ and $T\notin \mathcal{S}$. Consequently, T ∈ Comp $\left(K,\mathcal{S}\right)$ with T ∩ L an (n - i)-face, which leads to a contradiction by an inductive step.

It follows that for any two K, $L\in \mathcal{R}$ we have L ∈ Comp $\left(K,\mathcal{S}\right)$. Consequently, $\mathcal{R}=\mathsf{\text{Comp (}}K\mathsf{\text{,}}\mathcal{S}\mathsf{\text{)}}$ for some K such that $K\cap I_i^{+}\ne \varnothing$ and Comp ${\left(K,\mathcal{S}\right)}^{*}\cap I_i^{-}\ne \varnothing$. A contradiction that completes the proof.   □

A collection of cubes in $\mathcal{K}$ and the collection of the faces of all dimensions of cubes in $\mathcal{K}$ forms a polyhedral complex with $\mathcal{K}$ as its generating collection. Denote this complex by $\mathcal{X}$.

Suppose $f:\mathcal{V}\left(\mathcal{X}\right)\to ℝ$ is such that f (v) ≥ 0 for each $v\in \mathcal{V}\left(\mathcal{X}\right)\cap I_1^{+}$ and f (v) ≤ 0 for each $v\in \mathcal{V}\left(\mathcal{X}\right)\cap I_i^{-}$. We will make use of Theorems 3.1 and 4.1 to obtain an algorithm finding $\mathcal{C}\left(d\right)\subset \mathcal{K}$, for all $d\in {\mathcal{E}}_f$, such that the following is true

• 1. each $C\in \mathcal{C}\left(d\right)$ contains an edge from ${\mathcal{E}}_f$,

• 2. $\mathcal{C}{\left(d\right)}^{*}$ is connected, and

• 3. $\mathcal{C}{\left(d\right)}^{*}$ separates Iⁿ between $x\in I_1^{+}$ and $y\in I_1^{-}$ for each $x,y\in I^n\backslash \mathcal{C}{\left(d\right)}^{*}$.

Set $\mathcal{L}=\left\{\left[\frac{i}{k},\frac{i+1}{k}\right]\times \left\{0\right\}\times ...\times \left\{0\right\}:i=0,...,k-1\right\}$, and notice that ${\mathcal{L}}^{*}$is a segment joining $I_i^{+}$ and $I_i^{-}$. Therefore, $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$ will be the desired collection satisfying (1) - (3) for some $d\in \mathcal{L}$.

Algorithm (outline)

Step 1. Add all elements of ${\mathcal{E}}_f\cap \mathcal{L}$ to List A.

Step 2. Repeat Step 3-Step 11 until List A is empty.

Step 3. Pick an edge d from List A.

Step 4. Generate $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$. Remove d from List A.

Step 5. Add all elements $K\in \mathcal{K}$ such that $K\cap I_i^{+}\ne \varnothing$ to List B.

Step 6. Repeat Step 7-Step 9 until List B is empty.

Step 7. Pick a cube K from List B.

Step 8. Generate Comp (K, $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$). Remove K from List B.

Step 9. If there is L ∈ Comp (K, $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$) such that $L\cap I_i^{-}\ne \varnothing$ then go back to Step 3.

Otherwise, go back to Step 7.

Step 10. List all elements from $\mathcal{C}\mathsf{\text{(}}d\mathsf{\text{)}}$ ($\mathcal{C}{\left(d\right)}^{*}$ is a separator).

Step 11. Go back to Step 3.

Endnote

^aBy a map we will always mean a continuous function. Whenever continuity is not assumed we will use the term function instead.