A cubic graph is a graph whose vertices have degree three and a -graph is a graph whose vertices have degree either one or three. The graphs are allowed to have loops and parallel edges. Motivated by a result of Mochizuki , Liu and Osserman  associated a polytope to each -graph and studied its Ehrhart quasi-polynomial.
For each degree three vertex of a -graph , let , , and be the three edges incident to . Denote by the linear system consisting of a perimeter inequality and three metric inequalities defined on the variables , , and as follows:
From the three metric inequalities, one can immediately conclude that , , and are nonnegative. Consider the union of all the linear systems , taken over all degree three vertices of . The polytope is defined by the set of all real solutions for this linear system.
Let and be a weight function defined on the edges of
. We use the vector notation. In particular, when is a solution for the linear system defining , we write .
Given a rational polytope , Eugène Ehrhart defined the function , which is the number of lattice points in the closed dilated polytope , for a nonnegative integer parameter . Ehrhart showed that this function is a polynomial in when is an integral polytope. More generally, if is a rational polytope, the function is a quasi-polynomial whose period is closely related to the denominators appearing in the coordinates of the vertices of [6, 2].
Liu and Osserman conjectured ([10, Conj. 4.2]) that polytopes associated to connected -graphs with the same number of vertices and edges have the same
Ehrhart quasi-polynomial. They partially proved their conjecture, by showing that these quasi-polynomials coincide for all nonnegative odd values of the dilation parameter. In 2013, Wakabayashi [15, Thm. A(ii)] proved their conjecture.
An ingredient in Wakabayashi’s proof for Liu and Osserman’s conjecture is a local transformation performed in -graphs. This transformation is called an -move by Wakabayashi and it is also known as a nearest neighbor interchange (NNI). The NNI has been studied mainly for binary trees [4, 12], cubic graphs , and -graphs . We present the following general result for connected graphs with the same degree sequence, which might of interest on its own. We refer to a vertex of degree one in a graph simply as a leaf. An edge is external if it is incident to a leaf, otherwise it is internal.
Let and be connected graphs with the same degree sequence and the same set of external edges. Then
can be transformed into through a series of NNI moves.
One can choose a spanning tree in and a spanning tree in and require that all the pivots of the NNI moves are internal edges of both of these spanning trees.
One of the concerns of this paper is on the scissors congruence conjecture for the unimodular group, which is an analogue of Hilbert’s third problem (equidecomposability). Concretely, this was stated as the following question by Haase and McAllister . An integral matrix is unimodular if it has determinant . An affine unimodular transformation is defined by , where is a unimodular matrix and is a real vector.
[7, Question 4.1] Suppose that and are polytopes with the same Ehrhart quasi-polynomial. Is it true that there is a decomposition of into relatively open simplices and affine unimodular transformations such that is the disjoint union of ?
We show that for polytopes associated to -graphs such a scissors congruence decomposition holds. Namely, we have the following.
Let and be two connected -graphs with the same number of vertices and edges. Then there is a dissection of into smaller polytopes and affine unimodular transformations such that is the union of .
The proof of Theorem 3 relies on a piecewise unimodular transformation associated to a weighted version of the NNI move.
The rational polytope , associated to a -graph , enjoys some fascinating symmetry. Linke  considered the extension of for all nonnegative real numbers . Royer  defined a polytope to be semi-reflexive if for every nonnegative real number . One can verify that is semi-reflexive. A polytope is reflexive if it is integral, the origin is in its interior, and it is semi-reflexive . We prove the following.
For each -graph , the polytope is reflexive.
The paper is organized as follows. Section 2 contains the results involving the NNI move in graphs with the same degree sequence, including Theorem 1, while Section 3 discusses the extension of the NNI move to weighted graphs. Section 4 describes the unimodular decomposition of and presents the proof of Theorem 3. In Section 5, we prove Theorem 4. Finally, Section 6 contains some concluding remarks.
2 Nearest neighbor interchange
is an ordered pairconsisting of a set of vertices and a set , disjoint from , of edges, together with an incidence function that associates with each edge of an unordered pair of (not necessarily distinct) vertices of . The degree sequence of is the monotonic nonincreasing sequence consisting of the degrees of its vertices.
A nearest neighbor interchange (NNI) is a local move performed in on a trail of length three. This move interchanges one end of the two extreme edges of on the central edge (Figure 1). We refer to the central edge of as the pivot of the NNI move. The result of the move is another graph on the same number of connected components, with the same degree sequence. We consider the graph as having the same set of vertices and edges, that is, , and only the incidence function is adjusted accordingly. Intuitively, one can think of the edges as sticks that are being moved from to .
We think of an NNI move as a function that associates the graph to the graph . In symbols, . Observe that is also a trail in and .
The well-known rotation, used in data structures to balance binary trees, is a particular NNI move, performed on a -tree (Figure 2).
An NNI move does not affect the incidence to leaves, thus it preserves the partition of the edge set into internal and external edges.
Culik and Wood [4, Thm. 2.4] proved that any two -trees with (labelled) leaves can be transformed into one another through a finite series of NNI moves. They additionally gave an upper bound of on the number of NNI moves needed for this transformation. In this section, first we extend Culik and Wood’s theorem to trees with the same degree sequence (Lemma 8), which we then use to extend their result further to connected graphs with the same degree sequence (Theorem 1).
A caterpillar is a tree for which the removal of all leaves results in a path, called its central path, or in the empty graph. For the later, we define that the central path is empty.
Any tree can be transformed into a caterpillar with the same degree sequence through a series of NNI moves.
Let be a tree. If is a caterpillar, there is nothing to prove. So, we may assume is not a caterpillar. Let be a longest path in and an internal edge of not in such that is a vertex in . The vertex has two neighbors in , otherwise would not be a longest path, as is not in . Let and be the two neighbors of in . Let be a trail with edges , , and , where is a neighbor of other than . Perform an NNI on the trail as in Figure 3, to insert in , obtaining another tree with the same degree sequence and a path longer than . By repeating this process, we obtain a desired caterpillar after a finite number of NNI moves. ∎
The spine of a caterpillar is the sequence of the degrees of the vertices in the central path. We say the caterpillar is ordered if its spine is a monotonic nonincreasing sequence (Figure 4).
Any caterpillar can be transformed into an ordered caterpillar with the same degree sequence through a series of NNI moves.
An NNI can be used to swap any two adjacent vertices in the central path of a caterpillar, as in Figure 5(a). So we use an NNI to decrease, one by one, the number of inversions in the spine of a caterpillar until we obtain an ordered caterpillar. ∎
The external edges of a caterpillar can be sorted arbitrarily through a series of NNI moves.
An NNI can be used to swap any two external edges incident to two adjacent vertices in the central path of a caterpillar, as in Figure 5(b). ∎
Any two trees with the same degree sequence and the same set of external edges can be transformed into one another through a series of NNI moves.
Let and be two trees with the same degree sequence and the same set of external edges. Using Lemmas 5 and 6, we obtain a series of NNI moves that transforms into an ordered caterpillar with the same degree sequence and the same set of external edges of . Similarly, we obtain another series for . Using Lemma 7, we extend the series of NNI moves for to sort the external edges of the caterpillar coming from into the order they appear in the caterpillar coming from . The composition of these two series of NNI moves, with the series for inverted, gives a series of NNI moves that transforms into . ∎
Let be a connected graph that is not a tree, and let be an edge that is in a cycle. The graph obtained from by cutting is the graph resulting from the splitting of into two edges, each connecting one of the ends of to one of two new leaves (Figure 6).
Proof of Theorem 1.
Let be the number of vertices and be the number edges in and . The proof of the theorem is by induction on the dimension of the cycle space of and . If , then and are trees and the theorem follows from Lemma 8. Hence we may assume that .
Choose an edge in a cycle of and an edge in a cycle of . Let us denote these two edges by . Let and be the graphs obtained from and , respectively, by cutting their edge . The number of vertices in and is and the number of edges is . Call and the two new edges in both and . Since is in a cycle in and in a cycle in , the graphs and are connected and the dimension of their cycle space is . Also, and have the same degree sequence and the same set of external edges. By induction, can be transformed into through a series of NNI moves. The same series of NNI moves transforms into . Indeed, the set of external edges in all graphs obtained during the application of this series of NNI moves, transforming into , contain and . Glue and into an edge in each of these graphs, obtaining a sequence of connected graphs that is the result of a series of NNI moves starting at and ending at . This ends the proof of (4).
Similarly, by induction in the cycle space of and , one can prove (b). Indeed, it is sufficient to choose in each step of the induction an edge to be cut which is not in either of the spanning trees. ∎
For the case of cubic graphs, the proof of Theorem 1 provides an alternative proof for a theorem by Tsukui [14, Thm. II], which refers to NNI moves as -transformations, where the ‘S’ stands for slide. For the slightly more general case of -graphs, the proof of Theorem 1 provides an alternative proof for a proposition by Wakabayashi [15, Prop. 6.2], which refers to NNI moves as -moves. Wakabayashi’s proof uses a topological pants decomposition for compact, oriented surfaces of finite genus.
Figure 7 shows a series of NNI moves from the complete graph to a tree with a loop added to each of its leaves.
3 Weighted NNIs for -graphs
To deal with weights on the edges of a -graph, we now enhance the NNI move. This was achieved by a bijection defined by Wakabayashi [15, Prop. 6.3].
Let be a -graph and be a weight function defined on the edges of . A weighted NNI is a local move performed in on a trail of length three induced by a NNI move in on . The result of the move is the graph obtained from by applying an NNI move on , and the weight function defined on the edges of as follows.
Let be the central edge of , that is, the pivot of the NNI move. Let and be the other edges in , and and be the remaining edges adjacent to , as depicted in Figure 8. Possibly , , , and are not pairwise distinct. The weight function is such that for every and
Note that, if is integer valued, then so is . Moreover, since pivots are always internal edges and an NNI move does not affect the partition of the edges into internal and external, a weighted NNI move may only change the weights of internal edges.
We think of a weighted NNI move as a function which extends the previously defined NNI move, . Note that , because and
Wakabayashi [15, Prop. 6.3] proved that if and only if for every integer , which implies Liu and Osserman’s conjecture. We observe that this holds also for every real , and state it below as we will use it in the next section.
Let be a -graph and be a weight function defined on the edges of . Let be a trail in of length three and suppose that . Then if and only if for every . ∎
Therefore there are distinct rational polytopes whose Ehrhart quasi-polynomials coincide for all real .
As a weighted NNI changes only the weight of the pivot, which is always an internal edge, the series of NNI moves from to changes only the weights of internal edges. In fact, by Theorem 1(b), one can choose a spanning tree in and a spanning tree in and require that the series of NNI moves uses as pivots only internal edges of and . As a consequence, the bijection from to keeps fixed the majority of the coordinates of the points. Namely, it changes only coordinates that correspond to internal edges of the chosen spanning trees. Formally, the latter discussion provides the following as a corollary of Theorem 1(b) and Lemma 9. Let and be a weight function defined on the edges of . If is a subset of , then the restriction of to is the function such that for all .
Let and be connected -graphs with the same number of vertices and on the same set of edges. Then there exists a bijection between and such that, for , we have that
where is the set of internal edges of arbitrary spanning trees in and .
4 Scissors congruence
Haase and McAllister  have raised Question 2 that can be thought of as an analogue of Hilbert’s third problem (equidecomposability) for the unimodular group. We show that for polytopes associated to -graphs a statement similar to Question 2 holds.
Let be a -graph with edge set . Let be a trail in of length three and suppose that . As argued ahead, the function
is associated to one or two hyperplanes inand two or four unimodular transformations. For short, let for every . Let , , , , and be as in Figure 8, with , , and being the edges of . Clearly is piecewise linear, namely, for , for every and
The hyperplanes associated to are and , which are either the same hyperplane (if or
) or two orthogonal hyperplanes. Moreover, the matrix that gives the linear transformation in each case is unimodular. Indeed, the matrix for case (1a
) is obtained from the identity matrix by substituting the row corresponding to the edgeby the row , the matrix for case (1b) is obtained from the identity matrix by substituting the row corresponding to the edge by the row , the matrix for case (1c) is obtained from the identity matrix by substituting the row corresponding to the edge by the row , and the matrix for case (1d) is obtained from the identity matrix by substituting the row corresponding to the edge by the row . Thus the determinant of each such matrix is always . Therefore each of them is unimodular.
Figure 9 shows an example with the two 3-regular graphs on two vertices and their polytopes. The two graphs differ by one NNI move. The function between the two polytopes in Figure 9 is defined by , , and
which is piecewise unimodular. In this case, only one hyperplane (the one containing the shaded triangle inside the polytopes, ) splits the polytopes into two, each part being unimodularly equivalent to one of the parts of the other polytope. The corresponding unimodular transformations are given by
Note that the polytope on the right side of Figure 9 has one more vertex than the other one, so that the combinatorial types of these piecewise unimodularly equivalent polytopes may differ. Observe how the extra vertex is “formed” when going from the polytope on the left to the one on the right, and how it “disappears” when going in the other direction. Also, the edge between the origin and the vertex in the polytope on the left is not an edge in the polytope on the right.
Proof of Theorem 3.
The proof is constructive. By Theorem 1, there exists a finite sequence of NNI moves that transforms to , say We shall explain the procedure for the first two NNI moves assuming that both underlying weighted NNI moves are associated to only one hyperplane and consequently two unimodular transformations. The other cases and the rest of the NNI moves follow in an analogous but more cumbersome fashion.
Let be the trail used in the first NNI move, that goes from to , and let , for , be the corresponding weighted NNI move. Let be the hyperplane associated to , and let , be the two unimodular transformations, one for each side of the hyperplane . The hyperplane dissects the polytope into two smaller polytopes, and , so that . We now have that . In words, we presented a dissection of into two smaller polytopes and two affine unimodular transformations and that, if applied to the two smaller polytopes, result in . Now we will proceed one more step, and present a dissection of into four smaller polytopes, and four affine unimodular transformations that, if applied to the four smaller polytopes, will result in (Figure 10).
Let be the trail used in the second NNI move, that goes from to , and let , for , be the corresponding weighted NNI move. Let be the hyperplane associated to , and let , be the two unimodular transformations, one for each side of the hyperplane . The hyperplane dissects the polytope into two smaller polytopes, and , so that .
Now, note that dissects and each into two smaller polytopes, obtaining the dissection
The latter naturally induces the following dissection:
So, if we let
This completes the proof for the two first weighted NNI moves assuming that both are associated to only one hyperplane and consequently two unimodular transformations.
For the remaining cases, whenever a weighted NNI move is associated to two hyperplanes (and consequently four unimodular transformations), the polytopes would be dissected into up to four smaller polytopes, but the process would be essentially the same. ∎
An equivalent way to define a reflexive polytope is to require that it is integral and has the hyperplane description for some integral matrix . Another equivalent definition of reflexivity is to say that a polytope is reflexive if and only if the origin is in and where is the interior of .
Proof of Theorem 4.
The polytope consists of the vectors satisfying
for each degree three vertex and edges , , and incident to