On existence of reporter strands in DNA-based graph structures

Abstract

Through self-assembly of branched junction molecules many different DNA structures (graphs) can be assembled. We show that every multigraph can be assembled by DNA such that there is a single strand that traces each edge in the graph at least once. This strand corresponds to a boundary component of a two-dimensional orientable surface that has the given graph as a deformation retract. This boundary component traverses every edge at least once, and it defines a circular path in the graph that “preserves the graph structure” and traverses each edge.

1. Introduction

In recent years DNA nanotechnology has produced a wide variety of structures including two-dimensional arrays (e.g. [10,16]), three-dimensional polyhedra [1,15,4] and arbitrary graphs [8,14]. A computing model using DNA self-assembly of three-dimensional structures, i.e., spatial graphs, is proposed in [6,8]. This model relies on the ability to identify self-assembled DNA structures that are results of the computation. For this purpose it is proposed that these structures are designed such that, after ligation, a single reporter strand traverses the molecular structure completely. Then, the structure resulting from the computation can be identified through the existence of such a reporter strand. The molecular construction of a graph is done such that edges are represented as duplex DNA molecules and the vertices as branched junction molecules; a branched junction is a DNA arrangement wherein three or more double helices flank a single point [13]. Therefore, a strand can traverse an edge once, or twice. If there is a single strand that traverses each edge twice, then the whole molecule representing a graph is a folded single circular molecule. For example, the DNA graph structure reported in [14] was designed to be a single stranded circular DNA molecule when denatured. However, it was proved in [7] that there are graphs such that no DNA assembly of these graphs could be obtained as a single stranded circular molecule (for example, the graph in Fig. 1).

Fig. 1.

The directed graph obtained from the graph in (a) by replacing each edge with two oppositely oriented arcs is depicted in (b). Up to cyclic permutation, there is only one Eulerian cycle: a₁b₂a₂b₁a₃b₃. Every thickened graph for the graph in (a) has three boundary components. One such thickened graph is depicted in (c). A surface with a boundary component tracing the Eulerian cycle is depicted in (d). It has only one boundary component and is homeomorphic to the closed 2-ball.

In this paper we show that, although there are graphs that cannot be constructed as a single circular DNA molecule, each graph can be designed in such a way that a single circular molecule traverses each edge of the graph at least once. Such a reporter molecule, or a portion of such a molecule, can then be used to identify the assembled molecular structure through gel electrophoresis, perhaps combined with enzymatic approaches. This method was employed to identify the DNA structure assembled as a result of a solution of an instance of a three-colorability problem [17].

Our proof of the main result uses the same model used in [7] which is based on notions of topological graph theory. For a given graph G, we consider compact orientable surfaces, called thickened graphs of G, that have G as a deformation retract. The boundary curves of a thickened graph of G correspond to circular DNA strands that assemble (hybridize) into the graph G. The main result of this paper may also be of interest in topological graph theory. We show that every connected multigraph, even if it is not Eulerian, is always “almost” Eulerian. A graph is Eulerian if there is a cycle that visits every edge exactly once. It is well known that a connected graph in which every vertex has en even degree is Eulerian [3]. Hence, an Eulerian graph can be embedded in a thickened graph that contains a boundary component that traverses every edge exactly once. A directed graph (digraph) obtained from an non-directed graph by replacing each edge with two oppositely oriented arcs is Eulerian, because the in-degree of every vertex equals the out-degree [3]. This Eulerian cycle traverses every edge twice in opposite direction. A thickened graph with only one boundary component has a path that corresponds to such an Eulerian cycle in the corresponding digraph. However, in general, such digraph Eulerian cycle does not necessarily correspond to a boundary component of a thickened graph for the given graph (see Fig. 1), and so, it may not correspond to a DNA strand in a DNA structure representing the graph. Here we show that every connected (multi)graph has a property that is somewhat in between Eulerian and digraph Eulerian, i.e., for every connected (multi)graph there is a thickened graph with a boundary component that traverses every edge. The paper consists of two major sections. In Section 2 we show that the problem can be reduced to graphs containing vertices of degree three only, and in Section 3 we show that every 3-valent (multi)graph has the desired property and therefore can be designed such that there is a reporter DNA strand that traverses every edge at least once.

2. Reduction to 3-valent Graphs

2.1. Background and relationship to Eulerian cycles and walks

A mathematical model for investigating the number of DNA strands needed in construction of a molecule representing a graph was proposed in [7]. There, some basic properties as well as the upper and lower bounds on the number of DNA strands needed for molecular assembly of a graph structure are also given. A key tool used in this study is the notion of a thickened graph. For the completeness of the paper we recall some of the definitions here.

A (multi)graph G is a structure given by a triple (V, E, t) where V denotes the set of vertices, E the set of edges and t : E → W, where W is a set of one and two-element subsets of V defines the endpoints of the edges. If x ∈ t(e), we say that x is incident to e. Note that we allow multiple edges and loops. If e, e′ ∈ E are such that t(e) = t(e′) = {x, y}, we say that e and e′ are parallel edges between vertices x and y. In the case when t(e) = {x}, we say that e is a loop at vertex x. When there is no ambiguity, we will use two element subsets of V to indicate edges through their incident vertices.

A path in G is a sequence of vertices and edges p = v₀e₁v₁ … v_k−1e_kv_k such that t(e_i) = {v_i−1, v_i} for all i = 1, …, k. We extend the function t to paths, such that t(p) = {v₀, v_k}. If e ∈ {e₁, …, e_k}, then we say that p traverses e. The cardinality of {j | e_j = e} is the number of times p traverses e. Assume p traverses e twice, e is not a loop, and i, j are such that e_i = e = e_j. If v_i = v_j−1, then p traverses e in opposite directions. An empty path is a path that consists of a single vertex. A path with no repeating edges such that v₀ = v_k is called a cycle. A graph in which for every two vertices v, v′ there is a path p such that t(p) = {v, v′} is called connected. In this paper all graphs considered are connected.

Each strand of DNA is oriented, chemically this is often denoted say 5′–3′. Since a double stranded DNA molecule appears as a double helix of two oppositely oriented hybridized strands, we model a double helix as a strip such that the rims of the strip represent the two strands of the double stranded DNA. Even number of half-turns on the double helix is represented with a flat strip, and odd number of half-turns is represented with a half-twisted strip. The direction of a rim (or a DNA strand) is indicated with an arrow. Therefore, the two rims of a strip modeling a DNA duplex have arrows pointing in opposite directions. A more complex DNA structure is obtained by hybridizing and then ligating junction molecules with duplex DNA [14,17]. This can be seen as “gluing” the strips together such that the orientation of the edges is preserved, resulting in a structure that represents a compact orientable surface with boundaries. The boundary components of such manifold, which are the subject of this paper, represent the DNA strands that form the structure.

A compact orientable surface (two-dimensional manifold) with boundaries in which a graph G (considered as a 1-complex) can be topologically embedded as a deformation retract is denoted with F (G) and is called a thickened graph of G. In other words, the graph G appears as a subset of the thickened graph F (G) and there is a continuous map from F (G) onto G such that the restriction of the map on G is the identity. We refer the reader to a general topology textbook such as [11, 12] for the topological notions used, such as a deformation retract. The portion of F (G) that maps onto an edge e under the corresponding deformation retract is called an edge ribbon for e.

The set of boundary components of F (G) is denoted with ∂F (G). Note that for a given graph G, there may be many non-homeomorphic thickened graphs, and hence the number of boundary components of a thickened graph of G may not be unique. Therefore, the notation F (G) simply indicates that we choose and fix one such surface. In [7], graphs in which all vertices have degree three were considered. There, it was proved that the number of components of ∂F (G) is bounded by the first Betti number from above. Moreover, for a given graph, the parity of the number of components, and therefore the number of DNA strands needed for assembly of G is always the same, either always odd or always even.

For a boundary component σ of ∂F (G) we say that σ traverses edge e if a portion of σ belongs to the edge ribbon for e. Note that the set of edges traversed by σ remains the same under all deformation retracts for a given thickened graph. However, the set of edges traversed by a boundary component may be different for different thickened graphs. We say that σ visits vertex v if v is incident to an edge traversed by σ.

Observe that if there is a boundary component in ∂F (G) that traverses every edge, then G has a path that traverses every edge once or twice. A boundary component cannot traverse an edge more than twice, and since F (G) is orientable, traversing an edges a second time must be done in an opposite orientation than the first.

We are interested in the existence of a single boundary component that traverses every edge at least once. Eulerian graphs are graphs that have thickened graphs with a boundary component that traverses each edge exactly once. Graphs that have more than two vertices with odd degrees are not Eulerian and do not contain an Eulerian path [3]. A boundary component of a thickened graph that traverses every edge exactly twice can be considered as an Eulerian path in a digraph obtained from the given graph by replacing each edge with oppositely oriented arcs. Although every digraph obtained from a multigraph by replacing each edge with oppositely oriented arcs has a directed Eulerian cycle, this cycle does not necessarily correspond to a boundary component of a corresponding thickened graph (see Fig. 1). For the purpose of identifying and characterizing a three-dimensional DNA graph structure we are interested in the case that falls intuitively in between these two, i.e., we are interested in the existence of a single strand that traverses every edge once or twice. This question is closely related also to the Chinese Postman Problem: given a graph G, determine the smallest closed path (circuit) that visits every edge. Such path is known as Eulerian walk [9]. It is known that a path is Eulerian walk if and only if each edge appears at most twice and no more than half of the edges of a cycle appear twice in the walk [5]. But, similarly as in the case of Eulerian cycle in a digraph, an Eulerian walk may not be a portion of a boundary component of a corresponding thickened graph (see Fig. 2). In both examples in Figs. 1 and 2 an Eulerian cycle and an Eulerian walk trace boundary components of thickened graphs that do not “preserve” the structure of the given graphs (i.e., the thickened graphs fail to have the given graphs as deformation retracts). A path corresponding to a boundary component that visits every edge at least once may not be part of neither the digraph Eulerian cycle, nor the Eulerian walk. We call the boundary component σ ∈ F (G) a reporter strand for G if σ traverses every edge in G. Our main result states that for every multigraph G there is a thickened graph F (G) which contains a reporter strand.

Fig. 2.

Every Euler walk of the graph in (a) traverses one of the edges twice. One Eulerian walk e₁e₂e₃e₃e₄e₅ is depicted in (b). A thickened graph with a boundary component tracing this Eulerian walk is depicted in (c). There are two types of thickened graphs for the graph in (a), with three boundary components as depicted in (d) and with one boundary component as depicted in (e). None of these thickened graphs contains a boundary component tracing an Eulerian walk.

A degree-one vertex is represented with DNA as a hairpin (a segment of DNA with a self-complementary sequence that is folded back on itself), and a degree-two vertex is simply a double stranded DNA molecule, hence, the number of components of F (G), as well as traversing of the corresponding edges, does not change by removal of vertices with degree one or two. The one-degree vertices, together with their incident edges, can be removed completely. Also, a two-degree vertex v and its two incident edges e and e′ with t(e) = {v, w} and t(e′) = {v, w′}, can be substituted by an edge connecting vertices w and w′. Such removal of two-degree vertices may introduce parallel edges between w and w′, and if w = w′, these new edges show up as loops. Therefore, in the sequel, all graphs are connected multigraphs with vertices of degree at least 3.

First, we show that the problem can be reduced to 3-valent graphs. In particular, we show that for every (multi)graph G there is a graph T_G, that consists of vertices of degree 3 only, such that G is a homomorphic image of T_G. Moreover, for each thickened graph F (T_G) there is a thickened graph F_T (G) with a map from ∂F (T_G) onto ∂F_T (G) such that the edges traversed by a given component σ in F (T_G) map precisely to the edges traversed by the corresponding image of σ in F_T (G). In other words, we reduce the problem to the graphs that consist of vertices of degree 3 only. Multigraphs that contain only vertices of degree 3 are called 3-valent graphs.

2.2. Three-degree Perturbations

Recall that a graph homomorphism φ : G → G′ from a graph G = (V, E, t) to a graph G′ = (V′, E′, t′) is a pair of functions (φ_v, φ_e) such that φ_v : V → V′ and φ_e : E → E′ ∪ V′ satisfying: for every edge e ∈ E, if φ_e(e) ∈ E′, then t′(φ_e(e)) = φ_v(t(e)), and if φ_e(e) = v′ ∈ V′ then φ_v(t(e)) = {v′}. Note that this definition of graph homomorphism allows some edges to collapse to a vertex. Therefore, paths may map onto paths with shorter length. In this sense, every graph can be mapped homomorphically onto a graph with a single vertex and no edges.

We say that φ is surjective (onto) if E′ ⊆ φ_e(E). We drop the subscripts and write φ for both functions φ_v and φ_e when the context determines which of these has been used.

Remark

Given two graph homomorphisms ${\phi }^{\prime }=({\phi }_v^{\prime },{\phi }_e^{\prime }):G\to G^{\prime }$ and ${\phi }^{″}=({\phi }_v^{″},{\phi }_e^{″}):G^{\prime }\to G^{″}$ we can construct a composition of graph homomorphisms φ = (φ_v, φ_e) : G → G″ such that ${\phi }_v={\phi }_v^{″}\circ {\phi }_v^{\prime }$ and φ_e : E → E″ ∪ V″ defined as: for ${\phi }_e^{\prime }(e)\in E^{\prime }$ then ${\phi }_e(e)={\phi }_e^{″}{\phi }_e^{\prime }(e)$, and if ${\phi }_e^{\prime }(e)\in V^{\prime }$ then ${\phi }_e(e)={\phi }_v^{″}{\phi }_e^{\prime }(e)$. It follows directly from the definition that a composition of graph homomorphisms is a graph homomorphism.

Definition 1

Let G = (V, E, t) be a graph and v ∈ V be a vertex in G with degree k > 3 without a loop. Let {e₁, …, e_k} be the set of edges incident to v, and denote with v_i the other vertex incident to e_i (i.e., v_i ∈ t(e_i)). A 3-degree perturbation at v for graph G is a graph G′ = (V′, E′, t′) obtained from G by addition of k − 3 new vertices, such that V′ = V ∪ {u₁, …, u_k−3} and the set of edges

$$E^{\prime }=E\backslash \{e_3,\dots ,e_k\}\cup \{f_1,\dots ,f_{k-3},e_3^{\prime },\dots ,e_k^{\prime }\}$$

satisfying

$$\begin{array}{l}{t}^{\prime }(f_j)=\{u_{j-1},u_j\},\text{for}j=1,\dots ,k-3,(\text{take}{u}_0=v)\\ t^{\prime }(e_j^{\prime })=\{v_j,u_{j-2}\}\text{for}j=3,\dots ,k,(\text{take}{u}_{k-2}=u_{k-3})\\ t^{\prime }(e)=t(e)\text{for}e\in E.\end{array}$$

A 3-degree perturbation at a vertex v with loops {g₁, …, g_s} is a 3-degree perturbation at v by setting g_i = e_2i−1 = e_2i (i = 1, …, s), and $g_i^{\prime }=e_{2i-1}^{\prime }=e_{2i}^{\prime }$ with $t(g_i^{\prime })=\{u_{2i-1},u_{2i}\}(i=2,\dots ,s)$ in the above definition. Note that if G′ is a 3-degree perturbation at a vertex v, then v, and all new vertices u₁, …, u_k−3, have degrees equal to 3 in G′. Therefore, any further 3-degree perturbation at a vertex for G′ can only be done at a vertex in V \ {v}. In other words, in any subsequent 3-degree perturbation of G′, the graph G′ locally remains unchanged at vertices v, u₁, …, u_k−3.

A 4-degree vertex is shown in Fig. 3 (a.1) and (c.1). Two 3-degree perturbations of a four-degree vertex are presented in Fig. 3 (b.1) and (d.1). Note that a 3-degree perturbation at a given vertex is not necessarily unique, as seen in Fig. 3. Hence there may be several non-isomorphic graphs that can represent a 3-degree perturbation of a given graph.

Fig. 3.

Different 3-degree perturbations of the same four-degree vertex (b.1) and (d.1). For given thickened graphs of the 3-degree perturbations (b.2), (b.3), (d.2) and (d.3), the corresponding thickened graphs (lifts) for the original four-degree vertex are depicted in (a.2), (a.3), (c.2) and (c.3) respectively.

Proposition 2

If G′ is a 3-degree perturbation at a vertex v of G, then G is a homomorphic image of G′.

Proof

We use the notation introduced in Definition 1 and define a graph homomorphism φ : G′ → G in the following way. For each v ∈ V and e ∈ E, φ(v) = v, φ(e) = e. For each new vertex in G′, u_j ∈ V′ \ V, φ(u_j) = v (j = 1, …, k − 3), and for each new edge in G′, φ(f_j) = v and $\phi (e_j^{\prime })=e_j$. By the definition of G′, the map φ is an onto graph homomorphism.

The homomorphism φ defined in the proof of Proposition 2 is called a vertex perturbation.

Definition 3

A perturbed 3-degree graph, denoted T_G, for a graph G = (V, E, t) is a graph obtained from G by successive 3-degree perturbations at every vertex of degree higher than 3.

At each vertex there may be several 3-degree perturbations, so, a perturbed 3-degree graph for a given graph is not necessarily unique. From the definition of perturbed vertices, it follows that each vertex in G of degree more than 3 is perturbed exactly once and T_G is 3-valent.

Corollary 2.1

Let T_G = (V_T, E_T, t_T) be a perturbed 3-degree graph of G = (V, E, t). Then G is a homomorphic image of T_G.

Proof

Let G = G₀, G₁, …, T_G = G_k be a sequence such that G_i is a 3-degree perturbation at a vertex of G_i−1 for all i = 1, …, k. Then by Proposition 2 there are graph homomorphisms φ_i : G_i → G_i−1. The composition φ₁ ∘ · · · ∘ φ_k = φ_T is a graph homomorphism from T_G onto G.

Notice that if G has a set of vertices V and V_T is the set of vertices for a perturbed 3-degree graph of G, then by the definition V ⊆ V_T. The homomorphism φ_T is such that φ_T|_V = 1_V and for each v with incident edges {e₁, …, e_k} in G and 3-degree perturbation at v with new vertices u₁, …, u_k−3 in T_G, we have φ_T (v) = v = φ_T (u₁) = φ_T (u₂) = · · · φ_T (u_k−3). Also, if e ∈ E is an edge in G with t(e) = {v, w} and e′ ∈ E_T an edge in T_G such that φ_T (e′) = e, then t_T (e′) = {u, u′} where u (i.e., u′) is either v (i.e., w) or is one of the new vertices obtained by a perturbation of v (w, respectively). Therefore φ_T (u) = v and φ_T (u′) = w. These observations provide the following lemma.

Lemma 2.1

Let T_G = (V_T, E_T, t_T) be a perturbed 3-degree graph of G = (V, E, t) and, φ_T : T_G → G be defined as in the proof of Corollary 2.1. Let E′ ⊆ E_T be the set of edges such that $E^{\prime }={\phi }_T^{-1}(E)$. Then the restriction φ_T|_E′ : E′ → E is one-to-one.

Definition 4

Let T_G be a perturbed 3-degree graph of G and let the sequence G = G₀, G₁, …, G_k = T_G be such that G_i is a 3-degree perturbation at a vertex of G_i−1 for all i = 1, …, k. Let φ_i : G_i → G_i−1 be the corresponding vertex perturbations. The homomorphism φ_T : T_G → G obtained as the composition φ_T = φ₁ ∘ · · · ∘ φ_k is called a perturbation map.

2.3. Lifting the Thickened Perturbed 3-degree Graph

Let F (G) be a thickened graph of G and consider an orientation of F (G). In this case all boundary components are oriented such that the orientations of the boundaries at each edge ribbon (i.e., thickened edge) are in opposite direction (see e.g. [2,12]). To a boundary component σ of F (G) we can associate a finite sequence of vertices and edges obtaining a cyclic path pσ = (w₀e₁w₁ · · · w_n−1e_nw₀)_σ where w_i are vertices visited by σ and e_i are the edges traversed by σ. The vertices and edges in p_σ are listed in order determined by the orientation. Each edge e_j is incident to vertices w_j−1 and w_j, (we take w_n = w₀). This sequence is unique up to a cyclic permutation of p_σ. In other words, following the orientation, the boundary component σ visits vertex w_i, next it traverses e_i+1 and the next vertex visited by σ is w_i+1 (where i + 1 is taken mod n).

Given the orientation , for each vertex v ∈ V of degree k, we can associate a permutation in the following way. We order the incident edges of v with e₁, …, e_k. If v has s loops, we list the first 2s edges such that e_2i−1 = e_2i is the ith loop (i = 1, …, s). For a boundary component σ of F (G) that visits v, according to the orientation of F (G), if σ traverses edge e_i, visits v and next traverses e_j visiting v_j, in other words e_ive_jv_j is a substring of p_σ, then, we set (e_i) = e_j. Note that locally, in the thickened graph, there are k portions of boundary components that meet at a vertex with degree k. Each boundary component that visits v traverses two edges incident to v, and each edge has to be traversed twice in opposite orientations by boundary components.

Lemma 2.2

If v is a vertex in G of degree k, then for every oriented thickened graph F (G), the permutation is a k-cycle.

Proof

If there is a proper subset S ⊊ {e₁, …, e_k} such that (S) = S then the boundary components traversing an edge e_i ∈ S visits v and then traverses e_j ∈ S. This means, every boundary component that visits v, either traverses two edges in S, or it does not traverse an edge in S. In other words, the boundary components do not ‘connect’ the edges in S with those not in S. This would imply that the number of ribbons visiting v in the thickened graph is only #S. Any deformation retract then, would map the ribbons corresponding to the edges in S to edges meeting into one point, and the other thickened edges corresponding to the rest of the edges incident to v would map to edges meeting at at least one more point (as if v is split into two vertices, one incident to edges in S and the other incident to edges not in S). Thus, must be a k-cycle.

The observation from Lemma 2.2 has a physical meaning when a DNA structure representing a graph is considered. In a structure where edges are represented with duplex DNA, in order to have a DNA graph structure hybridized and hold together, it is necessary to have exactly k strands that meet at a junction of k-armed molecule representing a k-degree vertex. The DNA strands have opposite orientation in a duplex, hence, we take that the orientation corresponds to the strand orientation leading from 5′ to 3′. Thus the requirement that is a k-cycle implies that the k-degree vertex in a DNA-based graph structure is obtained as a k-armed branched junction.

Given two adjacent vertices v and w with a common incident edge e, we define a composition of with denoted in the following way. Let {e, e₂, …, e_k} be the incident edges of v and {e, f₂, …, f_m} be the incident edges of w. The composition is a map from set {e₂, …, e_k, f₂, …, f_m} to itself such that

$${(vw)}^{\mathcal{O}}(x)=\{\begin{array}{ll}{v}^{\mathcal{O}}(x)\hfill & \text{if}x\in \{e_2,\dots ,e_k\}\text{and}{v}^{\mathcal{O}}(x)\ne e\hfill \\ w^{\mathcal{O}}(x)\hfill & \text{if}x\in \{f_2,\dots ,f_m\}\text{and}{f}^{\mathcal{O}}(x)\ne e\hfill \\ v^{\mathcal{O}}(e)\hfill & \text{if}x\in \{f_2,\dots ,f_m\}\text{and}{w}^{\mathcal{O}}(x)=e\hfill \\ w^{\mathcal{O}}(e)\hfill & \text{if}x\in \{e_2,\dots ,e_m\}\text{and}{v}^{\mathcal{O}}(x)=e.\hfill \end{array}$$

In other words, we can imagine a ‘circle’ surrounding vertices v and w as if the edge between them is shrunken, the vertices v and w identified, and edges {e₂, …, e_k, f₂, …, f_m} are “sticking” out as being incident to the ‘circle’. The composition maps the set of incident edges to the ‘circle’ bijectively to itself according to the connection determined by the boundary components visiting v and w. A similar argument as the one in Lemma 2.2 shows that the map is again an s-cycle, where s = k + m − 2, since otherwise, the thickened graph would not be properly connected around vertices v and w preserving the graph structure.

Construction of the lift of a thickened perturbed 3-degree graph of G

Let G = (V, E, t) be a graph with vertices of degree at least 3 and let T_G = (V_T, E_T, t_T) be a perturbed 3-degree graph of G with φ_T the corresponding perturbation map from T_G to G. Let F (T_G) be a thickened graph of T_G and an orientation of F (T_G), and let E′ be, as in Lemma 2.1, the set of all edges in T_G that are not collapsed to a vertex by the perturbation map φ_T. Consider a vertex v ∈ V, its incident edges e₁, …, e_k in G and its perturbation in T_G with additional vertices u₁, …, u_k−3. Let γ_v : {e₁, …, e_k} → {e₁, …, e_k} be the map obtained as the composition ${\gamma }_v={\phi }_T{\gamma }_T^v{\phi }_T{\mid }_{E^{\prime }}^{-1}$ where ${\gamma }_T^v={({vu}_1\cdots u_{k-3})}^{{\mathcal{O}}^{\prime }}$. By Lemma 2.1, ${\phi }_T{\phi }_T{\mid }_{E^{\prime }}^{-1}:E\to E$ is the identity and as a consequence of Lemma 2.2 we have that ${\gamma }_T^v$ is a k-cycle. Therefore γ_v is a k-cycle permutation on edges {e₁, …, e_k}. We design a thickened graph F_T (G) of G with orientation such that such that γ_v = in F_T (G).

The desired thickened graph F_T (G) can be obtained by thickening G at vertex v in the following way. (An example of the step-by-step process for designing F_T (G) for a 5-degree vertex is depicted in Fig. 4.) Each edge e_i in E is thickened and can be represented as a ribbon. We orient the two rims of such a thickened edge in opposite directions. Consider one of the rims of the ribbon corresponding to e_i incident to vertex v as an “original”, and the other rim of the ribbon as an “image” for the map γ_v. The “original” has an arrow (orientation) pointing towards v and the “image” has an arrow (orientation) pointing away from v. If γ_v (e_i) = e_j then we connect the “original” rim of the ribbon corresponding to e_i with the “image” rim of the ribbon corresponding to e_j. The desired thickened graph F_T (G) is obtained when such attachments of the ribbons are performed at every vertex of G. Since every ribbon of a thickened edge has oppositely oriented rims, F_T (G) is oriented. Moreover, by construction, γ_v = for every vertex v ∈ V. The thickened graph F_T (G) is called the lift of the thickened perturbed 3-degree graph F (T_G).

Fig. 4.

The process of obtaining the lift graph F_T (G) from the thickened graph F (T_G).

Consider the example when v is a 4-degree vertex in G as in Fig. 3(a.1). Two 3-perturbations of v are shown in Fig. 3(b.1) and (d.1). Portions of a thickened graph F (T_G) for the 3-perturbations of v, in the neighborhood of v are shown in Fig. 3 (b.2) and (b.3) for the perturbation (b.1) and in Fig. 3(d.2) and (d.3) for the perturbation (d.1). Although (b.1) and (d.1) are different perturbation cases, the thickened graphs (b.2) and (d.2) define the same permutation (1 2 3 4). Hence they define the same lift, locally, at vertex v as seen in Fig. 3 (a.2),(c.2). The permutations defined by the thickenings shown in (b.3) and (d.3) are γ_v = (1 4 3 2) for the case depicted in (b.3), and γ_v = (1 4 2 3) for the case depicted in (d.3). The corresponding lifts in F_T (G) are presented in Fig. 3(a.3) and (c.3) respectively.

Notation

Let σ′ be a boundary component of F (T_G) and let φ_T : T_G → G be the perturbation map. If p_σ′ = (w₀f₁ · · · w_n−1f_nw₀)_σ′ we write φ_T (p_σ′) = (φ_T(w₀)φ_T (f₁) · · · φ_T (f_n)φ_T(w₀)) such that in φ_T (p_σ′) a sequence of consecutive repetitions of the same vertex, say v, v, …, v, is written as only one appearance of the same vertex v.

Proposition 5

Let G be a connected multigraph and F (T_G) a thickened graph of a perturbed 3-degree graph T_G of G. Let F_T (G) be the lift of F (T_G). Then for every boundary component σ′ in ∂F (T_G) there is a boundary component σ in ∂F_T (G) such that φ_T (p_σ′) = p_σ where φ_T : T_G → G is the perturbation map. Moreover, this correspondence is one-to-one, i.e., if σ′ ≠ σ″ in ∂F (T_G) then ${\phi }_T(p_{\sigma }^{\prime })\ne {\phi }_T(p_{\sigma }^{″})$.

Proof

Notice that for any boundary component σ of a given thickened graph, if we set p_σ = (w₀e₁w₁ … w_n−1e_nw₀)_σ then (w₁ · · · w_n−1 w₀) (e₁) = e₁, and also, w_i · · · w_i+r−1) (e_i) = e_i+r. Consider a boundary component σ′ ∈ ∂F (T_G) and let p_σ′ = (x₀f₁ … x_m−1f_mx₀)_σ′. Then for each i = 1, …, m, traversing the edges f_i and f_i+1, we have $x_i^{\mathcal{O}}(f_i)=f_{i+1}$ (we take i + 1 mod m). Let $E^{\prime }={\phi }_T^{-1}(E)$, then by Lemma 2.1 the restriction φ_T |_E′ : E′ → E is a bijection. Let U = (x_if_i+1 · · · x_i+r−1) (r ≥ 1) be a portion of p_σ′ which contains edges in E_T \ E′ such that f_i, f_i+r ∈ E′. (Note, when r = 1, then U = x_i.) Then x_i, …, x_i+r−1 are some of the vertices in T_G obtained as a 3-perturbation of the same vertex v in V and hence φ_T (x_i) = · · · = φ_T (x_i+r−1) = v. We can rewrite p_σ′ as $p_{{\sigma }^{\prime }}=(U_0{f}_1^{\prime }{U}_1\cdots f_s^{\prime }{U}_0)$ for some s, such that $f_j^{\prime }\in E^{\prime }(j=1,\dots ,s)$ and none of the U_j contains an edge in E′. Having U_j = (x_if_i+1 · · · x_i+r−1), then $f_i=f_j^{\prime },f_{i+r}=f_{j+1}^{\prime }$ and so, ${(x_i\cdots x_{i+r-1})}^{\mathcal{O}}(f_j^{\prime })=f_{j+1}^{\prime }$, i.e., (x_i · · · x_i+r−1) (f_i) = f_i+r. But this means that ${\gamma }_T^v(f_i)=f_{i+r}$ and φ_T (x_i) = φ_T (f_i+1) = · · · = φ_T (x_i+r) = v. If the perturbation map φ_T maps ${\phi }_T(f_j^{\prime })={\phi }_T(f_i)=e_j$ and ${\phi }_T(f_{j+1}^{\prime })={\phi }_T(f_{j+r})=e_{j+1}$, then by construction of the lift F_T (G) we have that (e_j) = e_j+1.

Therefore, the thickened graph F_T (G) has a boundary component σ_j in ∂F_T (G) for each U_j such that ( $f_j^{\prime }{\phi }_T(U_j)f_{j+1}^{\prime }$) is a portion (substring) of p_σj, traversing the edge $f_j^{\prime }$, visiting vertex φ_T (U_j) and then traversing the edge $f_{j+1}^{\prime }$. In other words, if ${\phi }_T(f_j^{\prime }{U}_j{f}_{j+1}^{\prime }{U}_{j+1}{f}_{j+2}^{\prime })=e_j{v}_j{e}_{j+1}{v}_{j+1}{e}_{j+2}$, then $v_j^{\mathcal{O}}(e_j)=e_{j+1}$ and $v_{j+1}^{\mathcal{O}}(e_{j+1})=e_{j+2}$, i.e., (v_jv_j+1) (e_j) = e_j+2. Thus, the boundary components σ_j and σ_j+1 must coincide. It follows that all σ_j’s belong to the same component σ in ∂F_T (G). Now applying the perturbation map φ_T to vertices and edges of σ′ we obtain ${\phi }_T(p_{{\sigma }^{\prime }})=({\phi }_T(x_0){\phi }_T(f_1)\cdots {\phi }_T(x_0))=({\phi }_T(U_0){\phi }_T(f_1^{\prime })\cdots {\phi }_T(U_{s-1}){\phi }_T(f_s^{\prime }){\phi }_T(U_0))=(v_0{e}_1\cdots v_{s-1}{e}_s{v}_0)=p_{\sigma }$.

The boundary component σ′ is unique that determines σ, since each segment $f_j^{\prime }{U}_j{f}_{j+1}^{\prime }$ appearing as a portion of p_σ′ is a path in T_G with non-repeating vertices and edges, and φ_T maps the edges $f_j^{\prime }$ one-to-one.

The above proposition states that given a graph G and its thickened perturbed 3-degree graph F (T_G), the thickened graph F_T (G) which is the lift of F (T_G) has the same number of boundary components as F (T_G).

Corollary 2.2

Let G be a connected multigraph, F (T_G) be a thickened graph of a perturbed 3-degree graph T_G of G. If there is a boundary component σ′ ∈ ∂F (T_G) that traverses every edge of T_G, then there is a boundary component σ ∈ ∂F_T (G) in the lift of F (T_G) that traverses every edge of G.

Proof

The proof follows directly from Proposition 5 applying the perturbation map, i.e., if p_σ′ = (x₀f₁x₁ · · · f_nx₀) is such that E_T = {f₁, …, f_n} then since E = φ_T (E_T), every edge from E appears in the sequence of φ_T (p_σ′) = p_σ.

3. Existence of a Reporter Strand for a 3-valent Graph

In this section we show that given a 3-valent (multi)graph T there is always a thickened graph F (T) such that one of the boundary components of F (T) traverses every edge of T at least once.

Before we pursue the proof of the main theorem we observe that by changing connections of the boundary components at a 3-valent vertex, which subsequently changes the local permutation defined by the boundary components, the number of boundary components can be reduced. Out of the six permutations on three elements, only two are 3-cycles: τ₁ = (1 2 3) and τ₂ = (1 3 2). The boundary connections of thickened 3-valent graphs that correspond to these two 3-cycles are depicted in Fig. 5(a) and (b). As shown by Lemma 2 in [7], if there are three boundary components that visit a vertex, by changing the connections of the boundary components from one type to the other, as depicted in Fig. 5(c) and (d), the three boundary components that visit the vertex unify in a single component. Hence, the number of boundary components reduces by two. We call the process of changing connections of the boundary components at a 3-valent vertex corresponding from one of the cycles τ₁ or τ₂ to the other, an elementary boundary operation.

Fig. 5. Elementary boundary operation.

Two different connections of boundary components at a 3-valent vertex: (a) corresponds to permutation τ₁ = (1 2 3) and (b) corresponds to permutation τ₂ = (1 3 2). Three boundary components that visit a given vertex (c), after an elementary boundary operation the vertex is visited by a single boundary component (d).

We have the following observation:

Lemma 3.1

Let G be a 3-valent graph and T (G) a thickened graph of G. If T (G) has only two boundary components σ₁ and σ₂, then there is a thickened graph T̂ (G) obtained from T (G) by elementary boundary operations which has two boundary components σ̂₁ and σ̂₂ such that σ̂₁ traverses every edge of G.

Proof

Since T (G) has only two boundary components, at every vertex, the edges incident to that vertex satisfy one of the two cases: (a) all edges are traversed by a single component twice, or (b) one of the edges is traversed by a single boundary component twice, and the other two edges are traversed by both components. Assume the components are colored with red and blue, ‘red’ being σ₁ and ‘blue’ being σ₂. Vertices that have all their incident edges traversed by the red (blue) boundary twice are called ‘red’ (‘blue’). Vertices that have edges traversed by both boundaries but one of the edges is traversed by the red (blue) boundary twice, is called red_b (blue_r). Suppose there is a vertex v which is blue_r. If such a vertex does not exist, then, every edge in T (G) is traversed by σ₁, i.e., every vertex is either red or red_b and the lemma holds. Note that if there are blue vertices, since the graph is connected there is a path from a blue vertex to a red (or red_b), and such a path must pass through a blue_r vertex. The local situation at vertex v, which is blue_r, such that its incident edge e is visited twice by σ₂, is unique up to symmetry and is depicted in Fig. 6(a). Since e is traversed by σ₂ twice, if w is the other vertex incident to e, then w is either also blue_r, or it is blue. Given an orientation of T (G), we can assume that the blue component σ₂ ‘leaves’ e, visits w, traverses some edges, and comes back to e. By performing elementary boundary operation at vertex v (see Fig. 6(b)) the connections of the boundary components σ₁ and σ₂ change to boundary components σ̂₁ and σ̂₂ of a new thickened graph of G. Now the edge e becomes visited by the other, red, boundary component σ̂₁ twice (consider components σ̂₁ and σ̂₂ in Fig. 6). Hence, v from being a blue_r vertex changes into being a red_b vertex. Notice that if w were blue_r it is now red, and if it were blue, then it becomes either red, or, red_b. Moreover, all edges traversed by σ₂ between the two visits of w are now traversed by σ̂₁. Therefore, with elementary boundary operations at blue_r vertices, the number of blue and blue_r vertices reduces. Observe further that after such elementary boundary operation, the number of boundary components in the new thickened graph remains to be two. Since the graph is finite, after finite number of elementary boundary operations all vertices remaining in the graph will be either red or red_b.

Fig. 6.

Reduction of the number of edges traversed by σ₂ twice.

We now proceed with the main result showing that for every 3-valent graph there is a thickened graph containing a reporter strand.

Theorem 3.1

Given a connected 3-valent graph G = (V, E, t), there is a thickened graph F (G) and σ ∈ ∂F (G) such that p_σ = (v₀e₁v₂ · · · v_ne_nv₀) and {e₁, …, e_n} = E.

Proof

We present the proof by induction on the number of vertices in G. Note that every 3-valent graph has to have an even number of vertices [3]. There are two two-vertex 3-valent graphs as depicted in Fig. 7(a). The existence of the single reporter strand for these cases is presented in Fig. 7(b). Indicated in bold (red) is the boundary component of the thickened graph that traverses every edge at least once. In the case of the first graph containing three parallel edges, the thickened graph has only one boundary component. On the other side, the graph with two loops (Fig. 7(a) right), has three boundary components, one of which traverses every edge at least once.

Fig. 7.

(a) Two 3-valent graphs with 2 vertices. (b) Thickened graphs for the graphs in (a) containing reporter strands which are indicated in bold (red).

Assume that the theorem holds for all connected 3-valent graphs with at most 2n vertices and consider a connected 3-valent graph G with 2n + 2 vertices. There are several possibilities.

Case 1. The graph G has a parallel edge incident to vertices v and w (see Figs. 8–10)

Fig. 8.

(a) Vertices v and w with parallel edges, both connected to a single vertex x. (b) The graph G′ containing two vertices less than G. The two edges incident to x, v and x, w respectively are replaced with a loop. (c) Addition of vertices v, w and the parallel edges. This introduces another boundary component indicated with dotted (green) line. (d) The reporter strand in a thickened graph obtained after an elementary boundary operation at vertex v.

Fig. 10.

(a) Vertices v and w with parallel edges, connected to two vertices x and y. (b) A graph G′ containing two vertices less from G obtained by removal of v and w. The two edges, one incident to x and the other to y (adjacent to v and w respectively) are replaced by a single edge f. The boundary component σ traversing every edge at least once, traverses f twice. (c) Addition of two vertices v, w and parallel edges connecting them. The boundary component σ indicated in bold (red) transfers respectively traversing every edge of G at least once.

Either v and w are connected with the third edge to the same vertex or not. The case when both v and w are connected to a single vertex x is depicted in Fig. 8. If we remove v and w from G and the edges incident to x, v and x, w respectively are replaced with a single edge incident to x, then we obtain a graph G′ with 2n vertices and a loop at x. Let e be the edge incident to x which is not the loop. By the inductive hypothesis there is a thickened graph with a boundary component σ traversing every edge of G′ at least once. The loop at x is therefore traversed by σ as indicated in bold (red) in Fig. 8(b). Because every thickened graph is orientable, every thickened graph of G′ has two boundary components traversing the loop, one of which traverses e twice. A boundary component traversing e, visiting x and then traversing the loop, cannot traverse the loop twice due to orientation constraints; so it has to visit x, ‘exit’ the loop and traverse e twice, leaving the second rim of the loop ribbon as a second boundary component. Hence σ must be the boundary component that traverses e twice as indicated in bold (red) in Fig. 8(b).

One can obtain graph G from G′ by placing two vertices on the loop (v and w), and adding another incident edge to both v and w. Consider a thickened graph of G such that the boundary component σ remains traversing the edges obtained by the addition of the two vertices on the loop (Fig. 8(c)). There are two additional boundary components. One component traverses opposite to σ along edges incident to {x, v} and to {x, w} which corresponds to the second boundary component of the loop ribbon of T (G′) (indicated with dashed line (purple) in Fig. 8(c)). And another boundary component, opposite σ along the parallel edges between v and w (indicated with dotted (green) line in Fig. 8(c)) is created. As three distinct boundary components visit v (and w) we can apply an elementary boundary operation at vertex v as indicated in Fig. 8(d). This results in a thickened graph with a single boundary component traversing all edges incident to x, v, w. Because in the rest of the graph this component traverses the edges the same way as σ traverses the edges of G′, and σ traverses every edge of G′, the new component traverses every edge in G.

The case when vertices v and w are adjacent to two distinct vertices x and y are depicted in Figs. 9 and 10. Again, by removing v and w from G and replacing their incident edges to x and y respectively with a new edge f incident to x and y (i.e., t(f) = {x, y}), we obtain a graph G′ with 2n vertices. By the inductive hypothesis, there is a thickened graph T (G′) with a boundary component σ that traverses every edge at least once. The case when σ traverses f only once is depicted in Fig. 9 and the case when σ traverses f twice is depicted in Fig. 10. The graph G is obtained by introduction of vertices v and w to G′. The corresponding thickened graph for G is obtained by keeping the pathway of σ in T (G) unchanged. This is done as if the ribbon representing the thickened edge of f is split parallel to the rims in the middle, but remains unchanged at the ends. The positions where the split starts and ends are the positions of vertices v and w. The boundary component σ is depicted in bold (red) in Figs. 9 and 10. When σ traverses f only once, there is another boundary component traversing f indicated as dashed (green) in Fig. 9(b). Denote this component with σ′. Consider a thickened graph F (G) that naturally extends from F (G′). Both σ and σ′ transfer accordingly to boundary components σ̂ and σ̂′ in F (G). The introduction of vertices v and w on the edge f and the addition of another edge incident to v and w introduces another boundary component in F (G) as indicated with dotted (purple) in Fig. 9(c). Since σ traverses every edge in G′, the only edge not traversed by the corresponding boundary component σ̂ in F (G) is the parallel edge incident to v and w obtained as a result of the addition of v and w to G′. This edge is traversed by two boundary components, the component σ̂′ corresponding to σ′ in F (G) and the new boundary component δ introduced by the addition of v and w to G′. Therefore there are three boundary components traversing the edges incident to v. By performing an elementary boundary operation at v, we obtain a thickened graph where these three boundary components are reduced to one (indicated in bold (red) in Fig. 9(d)).

Fig. 9.

(a) Vertices v and w with parallel edges, connected to two vertices x and y. (b) A graph G′ containing two vertices less than G. Two of the edges, one incident to x, v and the other incident to y, w are replaced by a single edge f. The boundary component σ traversing every edge, traverses f only once. The other boundary component σ′ traversing f is indicated with dashed (green) line. (c) Addition of two vertices v, w and the the parallel edges. The dashed ‘green’ component transfers respectively. Another boundary component δ indicated as dotted (purple) along the parallel edges is introduced. (d) The reporter strand in a thickened graph obtained after an elementary boundary operation at vertex v.

In the case when σ traverses f twice, the corresponding thickened graph of G has a boundary component σ̂ traversing every edge at least once without any additional changes (Fig. 10(c)). In this case the boundary components σ′, σ̂′ in Fig. 9(b),(c) respectively, are in fact the same as σ, σ̂.

Case 2. There are no parallel edges in G

Consider a graph G′ with 2n vertices obtained from G in the following way. Assume v, w are vertices in G and if v and w are adjacent through edge e, let e_v, $e_v^{\prime }$ be the other two edges incident to v and e_w, $e_w^{\prime }$ be the other two edges incident to w. The graph G′ is obtained by removal of vertices v, w with the edge e and by replacing the edges e_v, $e_v^{\prime }$ with a single edge f_v and replacing the edges e_w, $e_w^{\prime }$ with an edge f_w (see top of Fig. 11). Note that through this process f_v and f_w may become parallel edges or loops. We call G′ inductive reduction of G. Next, given a thickened graph T (G′) of G′, the boundary components traversing f_v and f_w form strips along the edges (away from the vertices). There may be up to four different boundary components traversing f_v and f_w in T (G′). They are indicated with σ₁, σ₂, σ₃, σ₄ in Fig. 11 bottom left. The corresponding thickened graph T (G) of the given graph G is obtained by placing a new ribbon strip representing the thickened edge e connecting the middle portions of the strips along the edges f_v and f_w (see bottom of Fig. 11). The positions where the new ribbon connects edge ribbons f_v and f_w are the places where vertices v and w are added to G′ to obtain the graph G. The boundary components σ₁, …, σ₄ change into new boundary components σ̂₁, …, σ̂₄. We call such obtained thickened graph T (G) the inductive thickened graph of G. Note that the thickened graphs T (G) and T (G′) are identical except in the neighborhood of vertices v, w where they differ by an additional ribbon along edge e in T (G).

Fig. 11.

The inductive reduction G′ obtained from G by removal of two adjacent vertices (top). The inductive thickened graph T (G) obtained from a thickened graph T (G′) (bottom). The boundary components are indicated in bold (red).

By the inductive hypothesis, there is a thickened graph T (G′) with a boundary component σ that traverses every edge at least once. We have the following possibilities:

1. σ traverses both edges f_v and f_w once.

2. σ traverses one of the edges f_v, f_w once and traverses the other edge twice.

3. σ traverses both edges f_v and f_w twice.

(a) The boundary component σ of T (G′) traverses both edges f_v and f_w once

This is the case when in Fig. 11 bottom left one of the portions σ₁ or σ₂ equals to one of the portions σ₃ or σ₄. Without loss of generality, we can assume that σ₁ and σ₄ are part of the same boundary component σ. There are two additional possibilities, either σ₂ and σ₃ are also part of the same boundary component δ ≠ σ (see Fig. 12(a), the boundary component σ is indicated in bold (red) and δ in dashed (blue)) or they are parts of two distinct components δ and ξ (see Fig. 12(b), the two distinct components are indicated in dashed (blue) and dotted (green)).

Fig. 12.

Edges f_v and f_w are traversed by σ exactly once. (a) The case when edges f_v and f_w in T (G_′) are traversed by two boundary components and (b) when they are traversed by three boundary components. (c) The boundary components in the inductive thickened graph obtained from the one in (a). There are three boundary components visiting vertex w. An elementary boundary operation at w unifies all three boundary components into one. (d) The boundary components in the inductive thickened graph obtained from the one in (b). The addition of a new ribbon can be chosen to unify σ with ξ.

When σ₂ = σ₃ = δ, the addition of vertices v and w, and the edge e in the inductive thickened graph ‘cuts’ the boundary component δ into two distinct components δ̂ and ξ̂ (see Fig. 12(c)). With this addition there are no changes in the boundary component σ, which, according to the inductive hypothesis, traverses every edge of G except e. However, the edges incident to v and w are now traversed by three distinct boundary components σ, δ̂ and ξ̂. By performing an elementary boundary operation at a vertex, say w, (see Fig. 5(c,d)), all three boundary components unify and the resulting boundary component traverses every edge of G.

In the case when σ₂ is a portion of a boundary component δ, and σ₃ a portion of another boundary component ξ, the addition of new ribbon in the inductive thickened graph T (G) can be chosen to connect components σ and ξ into a single boundary component σ̂ as depicted in Fig. 12(d). The new boundary component σ̂ then, in addition of traversing every edge of G′, and consequently every edge of G distinct from e, it also traverses e twice. Therefore, σ traverses every edge in G.

(b) The boundary component σ traverses one of the edges f_v, f_w once and traverses the other edge twice

Without loss of generality we can assume that in T (G′) the portions σ₁, σ₃ and σ₄ (see Fig. 11 bottom left) are part of σ as depicted with bold (red) in Fig. 13(a), and σ₂ is a portion of another boundary component δ depicted in dashed (blue). By addition of vertices v, w and edge e, the new ribbon in the inductive thickened graph T (G) unifies components σ and δ in a single boundary component σ̂ as depicted in Fig. 13(c). As a result, all edges in G are traversed by σ̂, including the edge e which is traversed twice.

Fig. 13.

At least one of the edges f_v, f_w is traversed by σ twice. (a) The case when edges f_v and f_w in T (G_′) are traversed by two boundary components σ and δ. (b) The case when f_v and f_w are both traversed by σ twice. (c) The boundary components in the inductive thickened graph obtained from the one in (a). The addition of a new ribbon unifies σ with δ. (d) The boundary components in the inductive thickened graph obtained from the one in (b). The component σ is cut, and an additional component δ is created.

(c) The boundary component σ of T (G′) traverses both edges f_v and f_w twice

In this case all component portions σ₁, σ₂, σ₃, and σ₄ in Fig. 11 bottom left belong to the same boundary component σ. This is illustrated in Fig. 13(b) representing σ in bold (red). The addition of a ribbon connecting vertices v and w along the edge e in the inductive thickened graph T (G) ‘cuts’ the boundary component σ in two separate pieces σ̂ and δ, illustrated with bold (red, σ) and dashed (blue, δ) in Fig. 13(d). Although there is a portion of σ̂ that traverses the new edge e, the part of σ̂ that has been cut, now δ, is a separate boundary component from σ̂ and, for example, might be traversing some edges in G twice, and therefore, those edges are not traversed by σ̂ in T (G). If every edge traversed by δ is traversed by δ exactly once, then σ̂ traverses every edge in G.

If T (G) contains only two boundary components σ̂ and δ, then by Lemma 3.1, there is a thickened graph T̂ (G) which contains a boundary component traversing every edge of G.

Suppose that there are at least three boundary components in T (G). By the inductive hypothesis, every vertex in G is visited by one of the components σ̂ or δ and every edge in G is traversed by at least one of σ̂ or δ. This follows because δ coincides with a portion of σ in T (G′) and σ traverses all edges of the inductive reduction G′. Again there are two possibilities, (1) there is a vertex u in G such that the edges incident to u are traversed by three distinct boundary components, two of them being σ̂ and δ, or (2) every vertex is visited by σ̂ (and possibly some other components different than δ) or it is visited by δ (and also possibly by some other components different than σ̂) or it is visited by σ̂ and δ only. In the former case, case (1), the connection of the boundary components at u is as depicted in Fig. 5(c), two of the three boundary components being σ̂ and δ. So, an elementary boundary operation at u unifies all three components visiting u and therefore it unifies σ̂ and δ. Hence the resulting boundary component traverses every edge. In the latter case, we perform elementary boundary operations, similarly as in the proof of Lemma 3.1, to reduce the number of edges traversed by δ twice. By construction, both δ and σ̂ visit v. We follow δ along its orientation, say along the edge e_v in Fig. 13(d). Then δ traverses edges that are also traversed by σ̂ until it reaches a vertex, call it u, incident to an edge that is traversed twice by δ. This situation is the same as the one depicted in Fig. 6(a) with δ being represented as σ₂. By an elementary boundary operation at u, the boundary component δ is “shortened”, all edges incident to u are traversed by σ̂ and by the proof of Lemma 3.1, the number of edges traversed by δ twice is reduced (Fig. 6(b)). We continue tracing δ along its orientation, following edges that are traversed by both σ̂ and δ, and perform the same operation whenever we encounter a vertex with an edge traversed by δ twice. When there are no edges traversed by δ twice, the boundary component σ̂ traverses every edge.