Genus Ranges of 4-Regular Rigid Vertex Graphs

We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used in modeling homologous DNA recombination. We show that the genus ranges are sets of consecutive integers. For any positive integer $n$, there are graphs with $2n $ vertices that have genus range ${m,m+1,...,m'}$ for all $0\le m<m'\le n$, and there are graphs with $2n-1$ vertices with genus range ${m,m+1,...,m'}$ for all $0\le m<m'<n$ or $0<m<m'\le n$. Further, we show that for every $n$ there is $k<n$ such that ${h}$ is a genus range for graphs with $2n-1$ and $2n$ vertices for all $h\le k$. It is also shown that for every $n$, there is a graph with $2n$ vertices with genus range ${0,1,...,n}$, but there is no such a graph with $2n-1$ vertices.


Introduction
Genus of a graph (also known as minimum genus) is a well known notion in topological graph theory, and has been studied for a variety of graphs (e.g., [14]). It represents the minimum genus of a surface in which a graph can be embedded. In this paper we introduce the notion of genus range of a given graph as the set of all possible genera of surfaces in which the graph can be embedded cellularly.
Graphs with 1-valent and 4-valent rigid vertices, called assembly graphs, have been used to model homologous DNA rearrangements, in which each 4-valent vertex represents a recombination site [2,3]. Simple assembly graphs are graphs that can be specified by double occurrence words, or unsigned Gauss codes, and are closely related to virtual knot diagrams [13]. Spatial embeddings of these graphs could be seen as a physical representations of the molecules at the moment of rearrangement, hence it is of interest to study certain combinatorial properties and embeddings of a given graph that reflect questions from such biological processes. Genara of graphs seem to measure their spacial complexity and are related to paths in the graphs that model assembled DNA segments. Thus in this paper we study the genus ranges of assembly graphs. Two main questions are considered: (a) Characterize the sets of integers that appear as genus ranges of assembly graphs with n 4-valent vertices for each positive integer n. (b) Characterize the assembly graphs with a given set of genus range.
With this paper we provide some answers to these questions. In particular, we show that a genus range of an assembly graph is always a set of consecutive integers (Lemma 2.10). Further, we show (Theorem 6.1) that every set {m, m + 1, . . . , m ′ } for 0 ≤ m < m ′ ≤ n appears as a genus range of some simple assembly graph with 2n vertices. The same family, without the set {0, 1, . . . , n}, appears as a genus range of some simple assembly graph with 2n − 1 vertices. We also observe that no simple assembly graph with 2n − 1 vertices has genus range {0, . . . , n} (Lemma 3.9) nor genus range {n} (Lemma 3.10). We characterize the simple assembly graphs with genus range {0}.
The paper is organized as follows. In Section 2 we give the definitions of assembly graphs, genus, and ribbon graphs (thickened graphs) and preliminary observations. Boundary components of ribbon graphs are closely related to genera, so this section introduces the basic techniques used in our results based on estimating the number of boundary components of ribbon graphs. Section 2 also contains results of computer calculations and a histogram of genus range distributions for graphs with 7 and 8 vertices. The algorithm in the computer program is based on the procedure described in [8]. A partial order on the sets of genus ranges is also introduced. In Section 3 some properties of genus ranges are listed.. In particular, we show that no graph with 2n − 1 vertices can have genus range {0, . . . , n} nor {n}. In Section 4 families of graphs that achieve certain sets of genus ranges are constructed. We characterize the assembly graphs with genus range {0}, and give a family of graphs that has genus range {n}. Further we provide a family of graphs with 2n vertices that have genus range {0, 1, . . . , n}. In Section 5 we find the genus range of tangled cords, a special subfamily of assembly graphs. This family achieves the maximum genus range according to the partial order for graphs with odd number of vertices. The summary of results characterizing genus ranges is given by Theorem 6.1 in Section 6. We end the paper with some concluding remarks.

Terminology and Preliminaries
In this section, definitions of the concepts used in this paper are recalled, notations are established, and their basic properties are listed.

Double Occurrence Words and Assembly Graphs
A graph is a pair (V, E) consisting of the set V of vertices and the set E of edges. The endpoints of an edge are either a pair of vertices or a single vertex. In the latter case, the edge is called a loop. The degree of a vertex v is the number of edges incident to v (each loop is counted twice).
An assembly graph [2] Γ is a finite connected graph where all vertices are 4-valent rigid vertices or vertices of degree 1. A vertex of degree 1 is called an endpoint. In this paper, we focus on assembly graphs with 4-valent rigid vertices only (without endpoints). Such graphs are studied in knot theory, and their spatial embeddings are also called singular knots and links ( [4], for example).
The number of 4-valent vertices in Γ is called the size of Γ and is denoted by |Γ|. An assembly graph is called trivial if |Γ| = 0. Two assembly graphs are isomorphic if they are isomorphic as graphs and the graph isomorphism preserves the cyclic order of the edges incident to a vertex.
A path in an assembly graph is called a transverse path, or simply a transversal, if consecutive edges of the path are never neighbors with respect to their common incident vertex. For an assembly graph without endpoints, a transversal is considered as the image of a circle in the graph, where the circle goes through every vertex "straight". An assembly graph that has an Eulerian (visiting all edges) transversal is called a simple assembly graph. For a simple assembly graph, if a transversal is oriented, the graph is called oriented.
We note that in a simple assembly graph, if a vertex v is an endpoint of a loop e, then e must be a neighbor of itself.
Convention: In the rest of the paper unless otherwise stated, all graphs are simple assembly graphs without endpoints.
Let A be an alphabet set. A double-occurrence word (DOW) over A is a word which contains each symbol of A exactly 0 or 2 times. DOWs are also called (unsigned) Gauss codes in knot theory (see, for example, [13]). The reverse of a word w = a 1 · · · a k is w R = a k · · · a 1 . Two DOWs are called equivalent if one is obtained from the other by a sequence of the following three operations: (1) (bijective) renaming the symbols, (2) a cyclic permutation, (3) taking the reverse. For example, w = 123231 is equivalent to its reverse w R = 132321 and w ′ = 213132 is w R after renaming 1 with 2 and 2 with 1. Therefore, all these words are equivalent. If for a DOW w there is an equivalent DOW w ′ such that w ′ = uv, where u and v are two non-empty DOWs then w is called reducible; otherwise, it is called irreducible.
Double-occurrence words are related to assembly graphs as follows. Let Γ be an oriented simple assembly graph. Let the set of 4-valent vertices of Γ be V = {v 1 , . . . , v n } where n = |Γ|. Pick and fix a base point on the graph and an orientation of a transversal. Starting from the base point, travel along the transversal, and write down the sequence of vertices in the order they are encountered along the transversal. This gives rise to a DOW over alphabet V . Conversely, for a given DOW containing n letters, an assembly graph is constructed with n vertices (labeled with the letters) by tracing the labeled vertices in the order of their appearances in the DOW. It is known that equivalence classes of DOWs are in one-to-one correspondence with isomorphism classes of assembly graphs [2]. In particular if w is a DOW, we write Γ w for a simple assembly graph corresponding to w.

Genus of Simple Assembly Graphs
If a graph admits an embedding on the plane, it is called planar. An embedding of a graph in a surface is called cellular if each component of the complement of the graph in the surface is an open disk. For a graph G, the minimum orientable genus of G, denoted g min (G), is the smallest non-negative integer g such that G admits an embedding in a closed (compact, without boundary) orientable surface F of genus g. We recall the following fact (Proposition 3.4.1 in [14]): Every embedding of G into a minimum genus surface is cellular. The maximum orientable genus of G, denoted g max (G), is the largest non-negative integer g such that G admits a cellular embedding in a closed orientable surface F of genus g. In this work we are concerned only with embeddings in orientable surfaces. When an assembly graph is embedded in an orientable surface, we assume that the cyclic order of edges at each vertex agrees with the embedding. In particular, neighboring edges at a vertex belong to the boundary of a common complementary region.
Definition 2.1. The genus range gr(Γ) of an assembly graph Γ is the set of values of genera over all surfaces F into which Γ is embedded cellularly. We denote the family of all genus ranges of assembly graphs with n vertices by GR n .
One way to obtain a cellular embedding of an assembly graph Γ in a compact orientable surface is by constructing a surface by connecting bands (ribbons) along the graph. This construction is called a ribbon construction, and is used in estimating the number of DNA strands in DNA molecules representing graph structures [11,12]. The construction is outlined below.
Let Γ be an assembly graph with n vertices labeled from 1 to n, and let w be the DOW that represents Γ with respect to a transversal. To each vertex i, i = 1, . . . , n, we associate a square with coordinate axes coincide with the edges incident to the vertex as depicted in Figure 1. Each side of the square corresponds to an edge incident to the vertex such that neighboring sides of the square that correspond to neighboring edges. For an edge e with the end points i and j, we connect the sides of the squares at i and j corresponding to e by a band. The bands are attached in such a way that the resulting surface is orientable. In Figure 1(A), the connection by a band is described when the vertex j immediately follows i in w, where j is the first occurrence  Figure 1(B), one possibility of connecting a band to the vertex j at its second occurrence is shown. By continuing the band attachment along a transversal, one obtains a compact surface with boundary. The resulting surface is called a surface obtained from ribbons, obtained by the ribbon construction, or simply a ribbon graph. This notion has been studied in literature for general graphs (see, for example, [10]). By capping the boundaries by disks, one obtains a cellular embedding of Γ in an orientable, closed (without boundary) surface. For a given cellular embedding of an assembly graph Γ, its neighborhood is regarded as a surface obtained from ribbons as described above. Note that there are two choices in connecting a band to a vertex j at its second occurrence, either from the top as in Figure 1(B), or from the bottom. Hence 2 n ribbon graphs can be constructed that correspond to all cellular embeddings of a given assembly graph. Therefore, the genus range of a given assembly graph can be computed by finding genera of all surfaces constructed from ribbons. In [6,7], the two possibility of connecting bands are represented by signs (±), and signed Gauss codes were used to specify the two choices.

Computer Calculations
In this section we summarize our computer calculations and observations. The calculations help understand the structure of genus ranges and formulate conjectures. Calculations are based on a description of boundary curves of ribbon graphs in [8].    To put these calculations into perspective, we define the notation [a, b] for the set {a, . . . , b} for integers 0 ≤ a ≤ b. We define the consecutive power set of {0, 1, . . . , n} for a positive integer n, denoted by CP(n), to be the set of all consecutive positive integers: We define the following linear order on consecutive power sets. Note that from the definition it follows that the partial order ≤ on CP(n) is a linear order for any positive integer n. [2,3] b a [2,2] ( [3,3]) The order relation [2,2] ≤ [1,3] in GR 5 is denoted by a dotted line, which is immediate predecessor/successor in GR 5 but not in CP (3).
The genus ranges GR 7 and GR 8 , as well as the number of graphs with each genus range arranged in the linear order are depicted in Figure 3.

Computing the Genus Range of an Assembly Graph
First we recall the well-known Euler characteristic formula, establishing the relation between the genus and the number of boundary components. The Euler characteristic χ(F ) of a compact orientable surface F of genus g(F ) and the number of boundary components b(F ) are related by χ(F ) = 2 − 2g(F ) − b(F ). As a complex, Γ is homotopic to a 1-complex with n vertices and 2n edges, and F is homotopic to such a 1-complex. Hence χ(F ) = n − 2n = −n. Thus we obtain the following well known formula, which we state as a lemma, as we will use it often in this paper. Thus we can compute the genus range from the set of the numbers of boundary components of each ribbon graph, { b(F ) | F is a ribbon graph of G }. Note that n and b(F ) have the same parity, as genera are integers. Next we compute the number of boundary components of ribbon graphs for a given assembly graph. In Figure 4(A), the boundary curves of a ribbon graph of an assembly graph, near a vertex, are indicated. The arrows of these boundary curves indicate orientations of the boundary components induced from a chosen orientation of the ribbon graph. If in Figure 1(B), the direction of entering the vertex has been changed from top to bottom (or vise versa), the ribbon graph changes. This change in the ribbon graph is illustrated in Figure 4(B). Note that the new ribbon graph is orientable, as indicated by the arrows on the new boundary components. We use a schematic image in Figure 4(C) to indicate the changes of connections of the boundary components illustrated in Figure 4(B). We call this operation a connection change. Thus starting from one ribbon graph for a given assembly graph Γ, one obtains its genus range by computing the number of boundary components for the surfaces obtained by switching connections at every vertex (2 n possibilities for a graph with |Γ| = n for a positive integer n). Example 2.9. In Figure 5, boundary curves of two ribbon graphs are shown for the graph representing the word 121323. In Figure 5(A), one sees that the boundary curve is connected, and by Lemma 2.8, its genus is 2. In Figure 5(B), where the connection at vertex 3 is changed, one sees that the boundary curves consist of 3 components, and hence its genus is 1. In fact we will find that the genus range of this graph is {1, 2} in Section 5.
In Figure 6, all possibilities of global connections of local arcs for a neighborhood of a vertex corresponding to Figure 4(A) are depicted in the top row, and in the middle row, the connections after the change of connection as in Figure 4(C) are depicted. In the bottom row, the changes of the numbers of boundary components are listed. In Figure 7, we illustrate how all possible connections are obtained. We label the edges at a vertex by 1, 2, 3, 4 as in the left of the figure, following the cyclic order of the rigid vertex. Each edge has two boundary curves, also labeled by the same number, in opposite orientations. For a given ribbon graph, the boundary arcs of each edge are connected away from the neighborhood of the vertex. The connection is depicted by dotted lines. This connection by dotted lines defines a permutation of four letters as follows. In (b), for example, a dotted line connects the outgoing boundary curve of the edge 1 to the incoming curve of the same edge, so the permutation sends 1 to 1. The outgoing boundary curve of the edge 4 is connected by a dotted line to the incoming curve of the edge 3, so the permutation sends 4 to 3. Thus the connection (b) corresponds to the permutation (1243). By symmetry of 90 degree rotations, the same connection corresponds to three other permutations (1324), (2134), and (4231). Up to symmetry, the connection (d) corresponds to two permutations, and (f) corresponds to a single permutation. Each of the connections (b), (c), (e) and (g) in Figure 6 corresponds to four permutations, each of (d) and (h) corresponds to two, and each of (a), (f), (i) and (j) corresponds to only one permutation. This exhausts all 24 permutations.
From computer calculations we notice that the genus range always consists of consecutive integers. Indeed, we prove the following. Proof. Let Γ be an assembly graph and F be a ribbon graph. Let gr(Γ) be the genus range of Γ. and let a, b be the minimum and the maximum integers in gr(Γ), respectively. Let F, F ′ be the corresponding ribbon graphs with genus a and b, respectively. Then F ′ is obtained from F by changing the boundary connections at some of the vertices. Thus there is a sequence of ribbon graphs F = F 0 , F 1 , . . . , F k = F ′ such that F i+1 is obtained from F i by changing the boundary connection at a single vertex for i = 0, . . . , k − 1. When a connection is changed at one vertex, the number of boundary components changes by at most two as indicated in Fig. 6, where the change in the number of boundary components are indicated at the bottom of the figure. By Lemma 2.8, the difference of the genus between two consecutive ribbon graphs F i and F i+1 is at most one. Hence there exists a cellular embedding of Γ with genus c for any c ∈ Z such that a < c < b. Thus gr(Γ) consists of consecutive integers.

Properties of Genus Ranges
In this section we investigate properties of genus ranges that will be used in later sections. Although the following lemma is a special case of Lemma 3.6, we state it separately since its simple form is convenient to use.
Lemma 3.1. For any DOW w, the corresponding genus range of Γ w is equal to that of Γ w ′ where w ′ = waa and a is a letter that does not appear in w.
Proof. The assembly graph Γ w ′ corresponding to w ′ is obtained from the graph Γ w for w by adding a loop with a single vertex corresponding to the letter a. Both of the two boundary connections at the added vertex a increases the number of boundary components by one as depicted in Figure 8 i is a cyclic permutation or reverse of w i and a i is a single letter that does not appear in w i for i = 0, . . . , n − 1.
A DOW obtained from the empty word ǫ by loop nesting is called loop-nested. An assembly graph corresponding to a loop-nested DOW is called a loop-nested graph. Lemma 3.1 implies that loop nesting preserves the genus ranges. Specifically, Corollary 3.3. If a DOW w ′ is obtained from a DOW w by loop nesting, then their corresponding assembly graphs have the same genus range.
Corollary 3.4. If a set A appears as the genus range in GR n for a positive integer n, then A appears as a genus range in GR m for any integer m > n.
Proof. Let w ′ = wa 1 a 1 a 2 a 2 · · · a k a k where k = m − n > 0, then by Corollary 3.3 Γ w ′ has the same genus range as Γ w .
Definition 3.5. Let Γ 1 and Γ 2 be assembly graphs. An assembly graph Γ is said to be obtained from Γ 1 and Γ 2 by a cross sum if it is formed by connecting the two graphs to the figure-eight graph as depicted in Figure 9. This operation in relation to the number of boundary components was discussed in [9] in relation to virtual knots. The cross sum construction depends on the choice of edges of the two graphs to connect to the figure-eight. Note that the graph Γ obtained from Γ 1 and Γ 2 by a cross sum is reducible. Lemma 3.6. Let Γ 1 and Γ 2 be assembly graphs. If Γ is obtained from Γ 1 and Γ 2 by a cross sum, then Proof. For given ribbon graphs F 1 and F 2 of Γ 1 and Γ 2 , respectively, a ribbon graph F of Γ is constructed as depicted in Figure 10. The two boundary curves at the edge where Γ i (i = 1, 2) is connected to the figure-eight belong to either one component or two in each of F 1 and F 2 . In Figure 10(A), the case where the two curves belong to one component for both of Γ i , i = 1, 2, is depicted. In (B), the case when the two arcs belong to distinct components for one graph (Γ 2 in this figure) is depicted, and in (C), the case when the two arcs belong to distinct components for both of Γ i , i = 1, 2, is depicted. From the figures we identify these cases (A), (B), (C) with the connection cases (a), (b) and (h), respectively, in Figure 6. We see from Figure  6 that by changing the connections of the boundary components in all these cases, the number of boundary components does not change. Let v i and b i be the numbers of vertices and boundary components of F i , i = 1, 2, respectively, and similarly v, b for F . Then for all cases (A), (B) and (C) we have v = v 1 + v 2 + 1 and b = b 1 + b 2 − 1 and from Lemma 2.8 we compute as desired.   (i) Consider a ribbon graph F for Γ with genus g ∈ gr(Γ). For each ribbon graph F 0 in (A) through (D) of Γ 0 in Figure 12, we obtain a ribbon graph F ′ of Γ ′ by connecting corresponding boundaries of the ribbon graphs for Γ 0 and Γ. If the connection of the boundary curves of a ribbon graph F 0 of Γ 0 is as in Figure  12(E), then the two curves tracing e become connected, and there is an additional component indicated by a small circle in the middle of (E). So the number of boundary components remains unchanged, and the number of vertices increases by 2. Hence g(F ′ ) = g + 1, and g + 1 is an element of gr(Γ ′ ). If the connection of the boundary curves of a ribbon graph F 0 of Γ 0 is as in Figure 12(F), then again the number of boundary curves of the ribbon graph for Γ ′ remains the same as F , and the number of vertices increases by 2, so that g(F ′ ) = g + 1, hence g + 1 is an element of gr(Γ ′ ). Therefore gr(Γ) = [m + 1, n + 1].
(ii) If the connection of the boundary curves of a ribbon graph F 0 of Γ 0 is as in Figure 12(E), then the single boundary component tracing e becomes disconnected, and there is an additional component in F ′ . Therefore the number of boundary components increases by 2, and the number of vertices increases by 2. Hence g(F ′ ) = g, and g is an element of gr(Γ ′ ). If the connection of the boundary curves of a ribbon graph F 0 of Γ 0 is as in Figure 12(F), then the number of boundary components of the ribbon graph for Γ ′ remains the same as that of F . In this case the number of vertices increases by 2, so g(F ′ ) = g + 1, and g + 1 is an element of gr(Γ ′ ). Therefore gr(Γ) = [m, n + 1].
As to the condition (i) in Lemma 3.7, we note the following. Proof. Fix a planar diagram of Γ, still denoted by Γ. The closure of a topological, thin neighborhood of Γ in the plane determines a ribbon graph F . Each boundary curve of F corresponds to a region of the complement of F (a connected component of R 2 \ F ) in the plane. In this construction, the connection of boundary curves at every vertex has the form in Figure 4(A). We assign checker-board "black" and "white" coloring to these regions (there is one, see [13] for example) such that regions bordering the same edge have distinct colors. The corresponding boundary curves of F inherit this coloring. Hence every edge is traced by boundary components of two distinct colors. Let B and W be (disjoint, non-empty) families of boundary curves colored black and white, respectively. Any ribbon graph F ′ of Γ is obtained from F by changing boundary connections at some vertices. Let v 1 , . . . , v k be a sequence of vertices at which connections are changed in this order to obtain F ′ from F , and let F i be the ribbon graph obtained from F by changing boundary connections at {v 1 , . . . , v i }, 1 ≤ i ≤ k. When we change the boundary connection at v 1 and obtain F 1 , two diagonal boundary curves of the same color become connected. This corresponds to the connection (f ) in Figure 6. In Figure 4(A), the curves of the pair 1 and 3, and the pair 2 and 4 are of the same colors. The connected curves inherit the color, and each component of the boundary curves of F 1 is colored black or white, and every edge of Γ is traced by two boundary components of F 1 , one white and one black. Let B 1 and W 1 be families of boundary curves of F 1 colored black and white, respectively. They are disjoint and non-empty. Inductively, suppose that F i has two families of disjoint and non-empty boundary curves B i and W i , and that every edge of F i is traced by curves from both families. At v i+1 , the boundary curves are as depicted in Figure 4(A), where the curves of the pair 1 and 3, and the pair 2 and 4 are of the same colors, respectively, and by changing the connection to (C), the curves of the same colors are affected diagonally. Then the boundary curves of F i+1 inherit the colors from those of F i to form two families B i+1 and W i+1 , for i = 1, . . . , k − 1, and every edge of F i+1 is traced by curves of distinct colors. This concludes that the boundary curves at each edge of F k have two distinct colors, and hence they belong to distinct boundary components. Proof. Suppose there is such an assembly graph Γ. Since its genus range contains 0, Γ is planar. By Lemma 3.8, every ribbon graph of Γ has more than one boundary component, and therefore, the genus cannot be n by Lemma 2.8. Proof. We observe that for every assembly graph there is a ribbon graph with more than one boundary component. Then the Lemma follows immediately. Let Γ be an assembly graph with k vertices. Starting from one vertex, we enumerate the edges of Γ along the transversal from 1 to 2k. Since the transversal consists of an Eulerian path in which two consecutive edges are not neighbors, every vertex in Γ is incident to two even numbered edges and two odd numbered edges. Moreover, the odd numbered (similarly, even numbered) edges incident to a common vertex are neighbors to each other. Let G be the subgraph of Γ that is induced by the odd numbered edges. This graph consists of k vertices and k edges, so it is a collection of cycles. Consider one cycle v 1 , v 2 , . . . , v s of G. Let e 1 , e 2 , . . . , e s be the edges of this cycle such that v i is incident to e i−1 and e i for i = 2, . . . , s, and v 1 is incident to e 1 and e s . Consider the ribbon graph F of Γ obtained in the following way. Choose an arbitrary connection of the boundary components at vertex v 1 . One of the boundary components at v 1 , call it δ, traces both e s and e 1 . Then at v 2 we choose a connection of the boundary components such that δ traces both e 1 and e 2 . One of the connections (A) or (C) in Figure 4 provides this condition. Inductively, suppose the boundary connections at v 1 , v 2 , . . . , v i have been chosen such that δ traces e s , e 1 , . . . , e i . Then at v i+1 we choose the boundary connection such that δ traces e i and e i+1 . For i = s, we have that δ traces all edges e 1 , . . . , e s of the cycle. Note that δ must trace e s only once, i.e., it 'closes' on itself, because the other boundary component tracing e s traces also an even-numbered edge incident to v 1 , the other neighbor of e s . Since δ traces edges only in G, the ribbon graph F must contain boundary components that trace the edges not in G, hence, F has more than one component.

Realizations of Genus Ranges
In this section we construct graphs that realize some desired sets of genus ranges.   The graph corresponding to any word obtained from w = 12 · · · n12 · · · n, for an odd integer n, by loop nesting has the genus range {0, 1}.
Proof. First we show that the graph Γ n corresponding to a word w = 12 · · · n12 · · · n has genus range {0, 1}. We start with the planar diagram depicted in Figure 14. Boundary curves of the ribbon graph obtained as a neighborhood of Γ n are depicted by dotted lines. When the connection of one vertex (the vertex 1 in Figure 15 left) is changed, both pairs of diagonal curves are connected from two components to one (see Figure 6(f)). Hence the number of boundary components reduce by 2, and the genus increases from 0 to 1 by Lemma 2.8.
After the change at the vertex 1, the boundary connection at every other vertex 2 through n is as depicted in Figure 6(h). Hence the number of the boundary components does not change by changing the boundary connections. After the second change of the boundary connection at the vertex i, the connection at every vertex remains the same as in Figure 6(h). This property holds for odd n. In Figure 15 middle and right, changes are made at vertices i = 2 and 3, respectively, after the change at the vertex 1.
The statement follows from Lemma 3.1.  Any DOW whose corresponding graph has genus range {0, 1} is obtained from the word w = 12 · · · n12 · · · n for an odd integer n by loop nesting.   Proof. Computer calculations show that there is a unique graph of 4 vertices corresponding to 12314324 that has genus range {0, 1, 2} as mentioned in Remark 2.4. In Figure 16, an assembly graph P 3 is depicted that has the genus range {0, 1, 2, 3}. To show this, we start with the planar diagram as depicted. We consider the ribbon graph that is a thin neighborhood of this planar diagram. For this ribbon graph, the boundary connection at every vertex appears as type (A) in Figure 4. The genus of this ribbon graph is 0. Each region of the diagram corresponds to a boundary curve of the corresponding ribbon graph. Four of such boundary curves (out of total 8) are depicted by dotted lines in the figure, for the regions labeled by a 1 , a 2 , b 1 , and b 2 .
We change the boundary connection at the vertex labeled by v 1 from type (A) in Figure 4 to type (C), where the connection is as indicated in Figure 6(f). Then as Figure 6(f) indicates, the curves a 1 and b 2 , a 2 and b 1 , are connected, respectively, reducing the number of boundary components by 2. This causes the increase of the genus of the new ribbon graph by 1 by Lemma 2.8, so that the new surface has genus 1. After the connection change at v 1 , the boundary connection at v 2 remains as in Figure 6(f), since the regions b 1 , b 2 , c 1 , c 2 are bounded by distinct curves. Thus by repeating this process with vertices v 2 and v 3 , we obtain ribbon graphs of genus 2 and 3, respectively. The number of boundary curves after performing the changes at v 1 , v 2 and v 3 is 2. The two components can be seen as a checkerboard coloring of the regions, and the two boundary curves near every edge belong to distinct components (cf. Lemma 3.8). This shows that the genus range for this graph is {0, 1, 2, 3}.
Next we describe the assembly graph P n for n > 3 as the combination of three subgraphs indicated in Figure 19 as follows. If n is odd, then the subgraph in the box (X) in Figure 19 is as depicted in Figure  17(odd), and if n is even, it is as depicted in Figure 17(even), respectively. The middle part in the box (Y) is as depicted in Figure 19. The subgraph in box (Z) in Figure 19 is either (down) or (up) of Figure  18, and the choice is determined by the number of vertices |P n | = 2n. Specifically, for integers k, P n has the following patterns in boxes (X) and (Z) in Figure 19, respectively: If n = 4k, 4k + 1, 4k + 2, 4k + 3, Figure 19: Inductive construction of P n respectively, the box (X) and (Z) are filled with (even) and (down), (odd) and (up), (even) and (up), (odd) and (down), respectively. Examples for n = 6 (even, up) and 7 (odd, down) are depicted in Figure 20 and 21, respectively. We generalize the argument for P 3 to P n inductively. We start with the ribbon graph that is a thin neighborhood of the planar diagram as described in Figure 19, where the boundary connection at every vertex is of type (A) in Figure 4. The connection of boundary curves is as Figure 6(f) at every vertex. We successively change boundary connections from type (A) to type (C) at a subset of vertices we describe below. With every such change, the number of boundary components of the corresponding ribbon graph reduces by 2, and hence the genus increases by 1. We start by changing the boundary connections for the vertices in the subgraph in box (X). For n odd, we change the boundary connection at vertices v 1 and v 2 in Figure 22(odd), and for n even, at vertices v 1 , v 2 and v 3 in Figure 22(even). The checkerboard coloring in Figure 22 indicates that the boundary curves of the regions of the same shading are connected.  Next we proceed to box (Y) inductively. In Figure 23, after the changes of the boundary connections at vertices v i and v i+1 in this order, the boundary curves are connected for the curves of the regions of the same shading in Figure 23. Note that the boundary curves of the four regions (a) through (d) in the figure belong to distinct components. Hence the connection changes at the vertices v i+2 and v i+3 in this order connects the curves (a) and (c), (b) and (d) to those two curves corresponding to the two shadings, respectively. Inductively, by changing boundary connections in all thickened vertices in Figure 19 box (Y), all curves bounding the regions in box (Y) are connected to two boundary components. The two components are indicated by the checkerboard shading.
We complete the induction in box (Z) as indicated in Figure 4. The last sequences of vertices where the boundary connections are changed from type (A) to (C) are depicted in the figure for each case (up) and (down) of Figure 18. Proof. This follows from Proposition 4.5 and Lemma 3.4.

Genus Range of the Tangled Cord
In this Section we find the genus range of a special family of assembly graphs called the tangled cords. In view of Problem 1.1, we prove that for all n the maximum genus range (at least for odd n) {⌊ n−1 2 ⌋, ⌊ n+1 2 ⌋} is achieved by the tangled cord with n vertices. The analysis of the genus range in this case uses a technique of addition and removal of a vertex.

Addition and Removal of a Vertex
Let Γ be an assembly graph and e, e ′ be two edges in Γ with endpoints v 1 , v 2 and v ′ 1 , v ′ 2 respectively. We consider the graph Γ split (e, e ′ ) obtained from Γ by adding 2-valent vertices v and v ′ on e and e ′ , respectively, splitting e and e ′ into two edges e 1 , e 2 and e ′ 1 , e ′ 2 as depicted in Figure 25(middle). The new edges e 1 , e 2 , e ′ 1 , e ′ 2 have end vertices respectively. We say that Γ ′ is obtained from Γ by addition of a vertex by crossing e, e ′ in cyclic order e 1 , e ′ 1 , e 2 , e ′ 2 (or simply by addition of a vertex when context allows) if V (Γ ′ ) = V (Γ) ∪ {w} and the edges of Γ ′ are obtained from the edges of Γ split (e, e ′ ) by identifying v and v ′ to a single vertex w with cyclic order of edges e 1 , e ′ 1 , e 2 , e ′ 2 as depicted in Figure 25 from left to right. The cyclic order of the edges at vertices v 1 , v 2 , v ′ 1 , v ′ 2 remains as in Γ such that the roles of e, e ′ are taken by the corresponding new edges. If Γ ′ is obtained from Γ by crossing e, e ′ in cyclic order e 1 , e ′ 1 , e 2 , e ′ 2 , we write Γ ′ = Γ(e 1 , e ′ 1 , e 2 , e ′ 2 ). In this case we also say that Γ is obtained from Γ(e 1 , e ′ 1 , e 2 , e ′ 2 ) by vertex removal. Note that the vertices v 1 , v 2 , v ′ 1 , v ′ 2 need not to be distinct. Let F be a ribbon graph of Γ and let Γ ′ be obtained from Γ by addition of a vertex w. A ribbon graph F ′ of Γ ′ whose boundary connections at every vertex v = w is the same as the the boundary connections of F at v is denoted F (w). Since there are two possible connections of the boundary components of F (w) at the new vertex w, the notation indicates one of those two choices.
All possible changes in the global connections of boundary components of F (w) at the new vertex w relative to the boundary components of the crossing edges that add the new vertex w are depicted in Figure  26. Note that the bottom row coincides with the top row of Figure 6.

Figure 26: Possibilities of strand connections
Recall that the boundary component δ of a ribbon graph Γ traces the edge e of Γ if the boundary of the ribbon that contains e is a portion of δ. There are at most two boundary components that can trace an edge, so there are at most four boundary components of a ribbon graph that trace two edges from Γ.
Lemma 5.1. Let Γ, Γ ′ be two assembly graphs such that Γ ′ = Γ(e 1 , e ′ 1 , e 2 , e ′ 2 ) by crossing edges e, e ′ and addition of a new vertex w. Suppose F and F (w) are ribbon graphs for Γ, Γ ′ as defined above with b, b ′ being the number of boundary components for F and F (w), respectively. Then we have the following: Cases (ii), (iii) and (iv) follow similarly.

Genus Range of the Tangled Cord T n
A tangled cord [5] with n vertices and 2n edges, denoted T n , is a special type of assembly graph of the form illustrated in Figure 27. The graph T n corresponds to the DOW 1213243 · · · (n − 1)(n − 2)n(n − 1)n.
Specifically, T 1 , T 2 and T 3 correspond to 11, 1212, and 121323 respectively, and the DOW of T n+1 is obtained from the DOW of T n by replacing the last letter (n) by the subword (n + 1)n(n + 1). Figure 27 shows the structure of the tangled cord. For vertices and edges of T n , we establish the following notation for the rest of the section. The edges of T n are enumerated as e 1 , . . . , e 2n as they are encountered by the transverse path given by the DOW, in this order. The adjacent edges to each vertex are listed below, in the cyclic order around the vertex as follows.
We prove that for each T n , if F is a ribbon graph for T n then it has either 1 or 3 boundary components (if n is odd) or it has 2 or 4 boundary components (if n is even). First we prove the following lemma: Lemma 5.2. Let F be a ribbon graph for T n (n ≥ 2) and suppose B is the set of boundary components that trace edges e 2n−1 and e 2n . Then every edge of T n is traced by a curve in B.
Proof. We use the following notation: for any set E of edges, let B(E) be the set of boundary components of the ribbon graph F that trace at least one edge in E. So B in the statement is written as B({e 2n−1 , e 2n }).
Observe that B has at most four elements as there are at most four boundary components tracing two edges. Now edges e 2n−1 and e 2n in T n are not neighbors, but they are consecutive edges of the transverse path defining T n . Hence all boundary components tracing e 2n−1 and e 2n (which are at most four) must trace their neighboring edges e 2n−2 and e 2n−3 (see Figure 4 As a consequence of Lemma 5.2, and the fact that the parity of the number of boundary components of the ribbon graph must match the parity of the vertices from Lemma 2.8, we have that if n is even then any ribbon graph of T n has 2 or 4 boundary components, and if n is odd, then any ribbon graph of T n has 1 or 3 boundary components. In the following we observe that all these situations appear. Proof. This lemma is a corollary of (and the proof of) Lemma 3.9 where it is observed that for every assembly graph there is a ribbon graph with more than one boundary component. Lemma 5.4. For every odd number n > 2, there is a ribbon graph F with a single boundary component tracing edges e 2n and e 2n−1 whose global connection is as in Figure 26(f ).
Proof. Since n is odd, the number of boundary components are either 1 or 3. The global connection of the boundary component in Figure 26(f) implies that the four possible boundary components tracing edges e 2n and e 2n−1 (indicated in Figure 28(a) with 1 through 4) are connected such that 1 is connected to 2, 2 connected to 3. If 3 is connected to 1, there would be two boundary components, hence 3 must be connected to 4, and back to 1. Therefore we only need to show that there is a ribbon graph where 1 is connected to 2 and 2 to 3. Suppose a ribbon graph corresponding to a fixed planar diagram in Figure 28(a) has boundary curves connected as type (A) of Figure 4 at vertices v n , v n−1 , . . . , v 5 and also at vertex v 1 . Then following the indicated arrows in Figure 28 It follows that the two curves "meet" at every other vertex they visit (with indices changing in gaps of 3), but if the two curves "meet" at v i , one of the curves traces edge e 2i and the other traces e 2i+1 . There are three possibilities for the vertex of the smallest index (greater than 2) for ( * ) and ( * * ) to "meet": at vertex v 3 , vertex v 4 and at vertex v 5 respectively. These situations are depicted in Figure 28(b-d, respectively) where the blue component indicates (*) and the red indicates (**).
The curves "meet" at vertex v 3 when n = 6k + 3 ( Figure 28(b)), in which case we consider the ribbon graph where the boundary curves are connected at vertex v 2 in type (C) in Figure 4, and at all other vertices the connections are of type (A). The curves "meet" at vertex v 4 when n = 6k + 1 (Figure 28(b)), in which case we consider the ribbon graph where the boundary curves are connected at vertex v 4 in type (C), and at all other vertices the connections are of type (A). The curves "meet" at vertex v 5 when n = 6k + 5 ( Figure  28(b)), in which case we consider the ribbon graph where the boundary curves are connected at vertex v 2 and v 3 in type (C), and at all other vertices the connections are of type (A). In all three cases we note that: (i) the (blue) boundary curve 1 that traces e 3 to v 1 and then e 2n , continues to "join" with boundary curve 2 and (ii) the (red) boundary curve 2 that traces the even edges e 6 , e 8 , e 10 , . . . , e 2n−2 , and then e 2n−1 "joins" with boundary curve 3 (Figure 28(a)). Since in all three cases situations (i) and (ii) appear, the global connection of the boundary components at edges e 2n−1 and e 2n are the same as in Figure 26(f).
The following theorem gives the final result about the genus range of T n . Proof. Recall that there are at most 4 boundary components by Lemma 5.2. By Lemmas 5.3, 5.4 for every odd n, there is a ribbon graph for T n with one boundary component and there is a ribbon graph for T n with three boundary components. We need to prove, for n even, that there are ribbon graphs with 2 and 4 boundary components. The graph T n+1 is obtained from T n by vertex addition by crossing e 2n−1 and e 2n . Suppose n ≥ 3 is odd. Then T n has ribbon graphs with one and three components. By Lemma 5.1(iii) by adding vertex v n+1 through crossing e 2n−1 and e 2n the three-component ribbon graph for T n becomes a ribbon graph with two boundary components for T n+1 . By Lemma 5.4 there is a ribbon graph for T n with a single boundary component, in which the global boundary connection traces edges e 2n−1 and e 2n as in Figure 26(f). Then after adding vertex v n+1 by crossing e 2n−1 and e 2n , this ribbon graph becomes a ribbon graph with four boundary components for T n+1 . By Lemma 2.8 the result follows.
Corollary 5.6. The maximum genus range of assembly graphs with 2n − 1 vertices is [n − 1, n] for any n ∈ N.
Proof. This follows from Lemma 3.10 and Theorem 5.5.
Conjecture 5.7. The maximum genus range of assembly graphs with 2n vertices is [n − 1, n] for any n ∈ N.
This conjecture follows from another conjecture: [n, n] / ∈ GR 2n for any n ∈ N, together with Corollary 2.11 and Theorem 5.5.

Towards Characterizing Genus Ranges
In this section we summarize our results and use inductive arguments to determine genus rages we can characterize at this point. To describe the genus ranges we can realize, we define the following integer sequence. For positive integers k, ℓ ∈ Z ≥0 , let φ(k, ℓ) = 7k + 3ℓ − 1. For any n ∈ N, define K n = max{k ∈ Z ≥0 | φ(k, ℓ) ≤ n}, L n = max{ℓ ∈ Z ≥0 | φ(K n , ℓ) ≤ n} and ψ n = 3K n + L n . For n = 100, for example, K n = 14, L n = 1 and Ψ n = 43. For small integers, one computes their values as follows.
Proof. The part (iii) is a restatement of Lemmas 3.9 and 3.10. We show (i) and (ii) by induction and constructions. Computer calculations show that the statements hold for n = 1, 2, 3 for GR 2n−1 and GR 2n .
First we focus on sets [a, b] with a < b. Assume for induction that GR 2k−1 contains all sets [a, b] ∈ CP(k), where a < b ≤ k, excluding [0, k]. By Lemma 3.4, all these sets are contained in GR 2k . Proposition 4.5 implies that [0, k] is contained in GR 2k and so is in GR 2k+1 . Hence all sets [a, b] ∈ CP(k), where a < b ≤ k for n ≤ k, are contained in GR 2k and GR 2k+1 .
Assume next that GR 2k contains all sets [a, b] ∈ CP(k), where a < b ≤ k, and also that GR 2k−1 contains all sets [a, b] ∈ CP(k), where a < b ≤ k, excluding [0, k]. By Lemma 3.4, all [a, b] ∈ CP(k), where a < b ≤ k, are contained in GR 2k+1 . We show that GR 2k+1 contains all [a, b] ∈ CP(k + 1), where a < b ≤ k + 1, excluding [0, k + 1]. Among these wanted, the following have not been covered by the induction assumption: [h, k + 1] for h = 1, . . . , k + 1 and [0, k]. There is an assembly graph with 2k + 1 vertices and genus range [0, k] by Proposition 4.6. Let Γ be an assembly graph with 2k − 1 vertices that have genus range [h, k] for a fixed h, where 1 ≤ h ≤ k − 1. The ribbon graph of Γ with the highest genus k has connected boundary component. Therefore the condition (ii) of Lemma 3.7 is satisfied with any edge of Γ. Hence by Lemma 3.7, there is an assembly graph with 2k + 1 vertices and with the genus range [h, k + 1]. Thus GR 2k+1 contains all [a, b] ∈ CP(k + 1), where a < b ≤ k + 1, excluding [0, k + 1].
Finally we show that GR n contains [h, h] for 0 ≤ h ≤ ψ n . LetΓ be the graph with 6 vertices and genus range {3} corresponding to the word 123245153646 mentioned in Remark 2.3, Item 5. Assume that GR n contains [h, h] for 0 ≤ h ≤ ψ n for n ≤ k. For n = k + 1, let m = K n + L n . Consider the assembly graph Γ n obtained by cross sum construction as depicted in Figure 13, where in the boxes B 1 , . . . , B Kn copies of the graphΓ are inserted, while in B Kn+1 , . . . , B Kn+Ln copies of the graph Γ 0 corresponding to the word 1212 are inserted. By Lemma 3.6, we have gr(Γ n ) = [ψ n , ψ n ], and it holds for n = k + 1.
Remark 6.2. From Theorem 6.1, we conjecture that for any m ∈ N, there is an integer Ψ m ≥ ψ m such that For small values of m ≤ 20, the conjecture holds with Ψ m = ψ m for m = 1, . . . , 7,9,13. At this time we are not able to determine if [5,5] is in GR m for m = 10, 11, [6,6] for m = 12, [7,7] for m = 14, 15, [8,8] for m = 16, 17, 18, and [9,9] for m = 18, 19. The conjecture is true if there is only one unknown entry, and therefore, the conjecture holds for all m ≤ 20 except m = 18, and m = 18 is the smallest number for which we do not know if the conjecture holds. If one finds [8,8] / ∈ GR 18 but [9,9] ∈ GR 18 , then this will provide a counterexample. For n = 100, for example, we do not know if [h, h] ∈ GR 100 for h = 43, . . . , 50. We know that [4,4] is not in GR 8 only by computer calculations searching through all assembly graphs with 8 vertices.

Concluding Remarks
For regular 4-valent rigid vertex graphs, we defined the genus range, that are genera of cellular embeddings. Some ranges are realized by using specific families of graphs, and by some operations applied on graphs of smaller sizes. A few ranges are shown to be not realized as genus range. We identified certain ranges of singletons [h, h] that remain unknown whether they can be genus ranges.
In Figure 3(right), we note that there are only 13 graphs among 65346 graphs of 8 vertices with genus range [0,4]. In Theorem 4.5, we constructed graphs with 2n vertices with genus range [0, n], and these graphs seem to be very rare. Also we notice that among 7 vertex graphs, there are only 2 graphs with genus range [3,3] (see Remark 2.3, Item 5 and 6). It is a curious fact that certain types of genus rages are very rare, and some are numerous.
We remark on possible relations and applications to the studies on DNA assembly. A mathematical model using assembly graphs for DNA recombination processes is proposed and studied in [2,3,5], for example. The molecular structure in space at the exact moment of recombination is modeled by assembly graphs, and the assembled gene is modeled by Hamiltonian polygonal paths, that are paths that make "90 • turn" at every rigid vertex, and visit every vertex exactly once. Polygonal paths have been also studied in graph theory as A-trails (for example, [1]). In [5], it was proved that the tangled cord, which achieves the maximal genus range for at least the graphs with odd number of vertices, has the largest number of Hamiltonian polygonal paths among all assembly graphs of the same number of vertices. If one follows a boundary curve of a ribbon graph at every vertex (see Figure 4), the consecutive edges traced by the boundary curve must be neighbors, so they form a polygonal path. Thus it is expected that smaller numbers of boundary components contribute to larger numbers of Hamiltonian polygonal paths. However, at this time we don't know the nature of the relationship between the genus range and the number of Hamiltonian polygonal paths.