*l*_{1}-Embeddability Under Gate-Sum Operation of Two *l*_{1}-Graphs

- School of Science, East China Jiaotong University, Nanchang, China

An *l*_{1}-graph is one in which the vertices can be labeled by binary vectors such that the Hamming distance between two binary addresses is, to scale, the distance in the graph between the corresponding vertices. This study was designed to determine whether the gate-sum operation can inherit the *l*_{1}-embeddability. The subgraph *H* of a graph *G* is called a gate subgraph if, for every vertex *v* ∈ *V*(*G*), there exists a vertex *x* ∈ *V*(*H*) such that for every vertex *u* of *H*, *x* lies on a shortest path from *v* to *u*. The graph *G* is defined as the *gate-sum* of two graphs *G*_{1} and *G*_{2} with respect to *H* if *H* is a gate subgraph of at least one of *G*_{1} and *G*_{2}, such that *G*_{1}∪*G*_{2} = *G*, *G*_{1}∩*G*_{2} = *H*, and both *G*_{1} and *G*_{2} are isometric subgraphs of *G*. In this article, we have shown that the gate-sum graph of two *l*_{1}-graphs is also an *l*_{1}-graph.

## 1. Introduction

A computer network is a group of computer systems and other computing hardware devices that are linked together through communication channels to facilitate communication and resource-sharing among a wide range of users. Networks are usually visualized as a graph, with the computers or devices being represented by vertices and the connections between vertices shown as edges. Graham and Pollak [1] were concerned with message switching in interconnected loops of computers, and they studied the problem of addressing graphs with a ternary alphabet {0, 1, δ} such that any graph may be addressed with an edge distance of unity for some address length *n*. Blake and Gilchrist [2] restricted attention to the binary alphabet. They formulated a routing algorithm for message switching in computer networks that simplifies the computation of the minimum-length path between any two vertices. An *l*_{1}*-graph* is one in which the vertices can be labeled by binary vectors such that the Hamming distance between two binary addresses is, to scale, the distance in the graph of corresponding vertices [3]. The graph operation can construct a new graph from a given graph, and some properties can be inherited under these operations. Our motivation for this study was to determine which operations can inherit the *l*_{1}-embeddability. Thus, the purpose of this work is to determine the *l*_{1}-embeddability of the gate-sum graph of two *l*_{1}-graphs.

Let *G* = (*V, E*) be a connected simple graph. The *distance* between two vertices *u* and *v* of *G*, denoted by *d*_{G}(*u, v*), is the length of a shortest *u*–*v* path in *G*. Then [*V*(*G*), *d*_{G}] is a graphic metric space associated with *G* [3]. A subgraph *H* of *G* is an *isometric subgraph* if *d*_{H}(*u, v*) = *d*_{G}(*u, v*) for any *u, v* ∈ *H*. A subgraph of *G* is *convex* if, for any two vertices, it includes all of the shortest paths between them. Obviously, a convex subgraph of *G* is an isometric subgraph. Let *S*⊂*V*(*G*) be any subset of vertices of *G*. The *induced subgraph* *G*[*S*] is the graph that has the vertex set *S* and the edge set consisting of all edges in *E* for which both ends are in *S* [4].

Bandelt and Chepoi [5] introduced the definition of a gate subgraph. A subgraph *H* of a graph *G* is a *gate subgraph* if, for every vertex *v* ∈ *V*(*G*), there exists a unique vertex *x* ∈ *V*(*H*) such that *x* lies on the shortest path between *v* and any vertex *u* ∈ *V*(*H*); *x* is called the *gate* of *v*. Hammack et al. [6] showed that a gate subgraph is convex, but that a convex subgraph may not be a gate subgraph. For example, each subgraph induced by the black vertices in Figures 1A,B is a convex subgraph in each graph. The subgraph shown in Figure 1A is a gate subgraph, whereas that in Figure 1B is not.

If *u* and *v* are two vertices of a path, the subsequence of this path starting with *u* and ending with *v* is the *segment* of this path from *u* to *v*. The *shortest path* *P*_{xy} is the path connecting *x* to *y* that has the fewest edges. Clearly, the segment of a shortest path is still a shortest path [7].

The *l*_{1}*-space* is the metric space of sequences whose series is absolutely convergent, denoted by (*X, d*_{1}). Thus, *X* is the set of all real sequences *x* = (*x*_{1}, *x*_{2}, …) such that $\sum _{k=1}^{\infty}\left|{x}_{k}\right|<\infty $, and the distance function is defined as ${d}_{1}(x,y)=\sum _{k=1}^{\infty}|{x}_{k}-{y}_{k}|$ for any *x, y* ∈ *X*. A graph *G* is an *l*_{1}-*graph* if (*V*(*G*), *d*_{G}) is isometrically embeddable into some *l*_{1}-space. That is, there is a distance-preserving mapping φ from *V*(*G*) into *X* such that *d*_{G}(*x, y*) = *d*_{1}(φ(*x*), φ(*y*)).

The *n-dimensional hypercube* *Q*_{n} is the graph whose vertices are ordered *n*-tuples of 0s and 1s, two vertices being joined if and only if they differ in exactly one coordinate.

Assouad and Deza [8] showed that a graph *G* is an *l*_{1}-graph if and only if *G* is *scale-*λ*-embeddable* into a hypercube *Q*_{n} for some positive integers λ and *n*, meaning that there exists a mapping ϕ:*V*(*G*) → *V*(*Q*_{n}) such that

λ·*d*_{G}(*x, y*) = *d*_{Qn}(ϕ(*x*), ϕ(*y*))

for any *x, y* ∈ *V*(*G*). The integer λ is the *scale* of *G*. The smallest such integer λ is called the *minimum scale* of *G*. According to Shpectorov [9], the minimum scale λ of *G* is equal to 1 or is even. In particular, if λ = 1, *G* is an isometric subgraph of *Q*_{n}, also called a *partial cube*.

Shpectorov [9] and Deza and Grishukhin [10] showed that a graph *G* is an *l*_{1}*-graph* if and only if it is an isometric subgraph of the Cartesian product of cocktail party graphs and half-cubes. The *cocktail party graph* *K*_{n × 2} is a complete multipartite graph with *n* parts, each of cardinality 2, which is equivalent to a complete graph *K*_{2n} deleting a perfect matching, as shown in Figure 2. The hypercube *Q*_{n} is a bipartite graph, and the *half-cube* $\frac{1}{2}{Q}_{n}$ is the graph defined on one of two parts of this hypercube, with two vertices being joined if the distance between them in *Q*_{n} is 2.

An *l*_{1}*-rigid* graph is an *l*_{1}-graph that essentially admits a unique *l*_{1}-embedding. Shpectorov [9] showed that every *l*_{1}-rigid graph *G* is an isometric subgraph of a half-cube. He also proved that every *l*_{1}-*rigid* graph has scale 1 or 2. Deza and Laurent [11] proved that the complete graph *K*_{n} (*n* ≥ 4) and the cocktail graph *K*_{n × 2} (*n* ≥ 4) are not *l*_{1}-rigid, where the variety of *l*_{1}-embeddings of *K*_{n × 2} all come from that of the complete graph *K*_{n}. The half-cube graph $\frac{1}{2}{Q}_{n}$ (*n* = 3, 4) is *l*_{1}-rigid. Hence, they claim that, if *G* is not *l*_{1}-rigid, the variety of its *l*_{1}-embeddings arises from that of the complete graph. Deza and Tuma [12] and Chepoi et al. [13] studied the forbidden subgraphs of an *l*_{1}-rigid graph. They determined that an *l*_{1}-graph is *l*_{1}-rigid if and only if it is *K*_{4}-free.

Deza and Laurent [11] proved that the graph obtained by identifying single vertices from two *l*_{1}-graphs is also an *l*_{1}-graph. Wang and Zhang [14] proved that the graph obtained by gluing two *l*_{1}-graphs along an edge is also an *l*_{1}-graph if at least one of the original graphs is bipartite. However, for two non-bipartite graphs, this is not always the case. They also determined that even for two bipartite *l*_{1}-graphs, gluing a convex subgraph cannot guarantee the *l*_{1}-embeddability of the obtained graph. Naturally, we wondered if this result could be generalized.

Suppose that *H*_{i} is a subgraph of *G*_{i}, *i* = 1, 2. If *H*_{1} is isomorphic to *H*_{2}, their vertices can be identified under some isomorphism as a new graph *H* such that the incidence relationship between vertices and edges remains. The resulting graph is called the *H**-sum* of *G*_{1} and *G*_{2}, denoted by *G*_{1}∪_{H}*G*_{2}. In particular, if *H* is a single vertex *v* or an edge *e* = *uv*, the *H*-sum is called the 1*-sum* or the 2*-sum*, denoted by *G*_{1}∪_{v}*G*_{2} and *G*_{1}∪_{uv}*G*_{2}, respectively. Additionally, if *G*_{1} and *G*_{2} are isometric in *G*_{1}∪_{H}*G*_{2}, and *H* is a gate subgraph of at least one of *G*_{1} and *G*_{2}, then *G*_{1}∪_{H}*G*_{2} is called a *gate-sum* of *G*_{1} and *G*_{2}, denoted by ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Both *G*_{1} and *G*_{2} are isometric subgraphs of *G*_{1}∪_{H}*G*_{2} if and only if *d*_{G1}(*x, y*) = *d*_{G2}(*x, y*) for any *x, y* ∈ *H*.

For example, see the graph in Figure 3, where the marked *K*_{4} is an isomorphic subgraph of *G*_{1} and *G*_{2}. The *K*_{4}-sum graph *G*_{1}∪_{K4}*G*_{2}, shown in Figure 3C, is obtained by identifying these two marked *K*_{4} as the same subgraph. In particular, in Figure 3B, the marked *K*_{4} is a gate subgraph of *G*_{2}. Obviously, both *G*_{1} and *G*_{2} are isometric subgraphs of *G*_{1}∪_{K4}*G*_{2}. Therefore, it can be seen as a gate-sum graph ${G}_{1}{\cup}_{{K}_{4}}^{g}{G}_{2}$ of *G*_{1} and *G*_{2} with respect to *K*_{4}.

**Figure 3**. The gate-sum graph ${G}_{1}{\cup}_{{K}_{4}}^{g}{G}_{2}$ of *G*_{1} and *G*_{2} with respect to *K*_{4}.

In this paper, we have shown that the gate-sum graph of two *l*_{1}-graphs *G*_{1} and *G*_{2} is also an *l*_{1}-graph. The remainder of this article is organized as follows. In section 2, we have introduced the concept of convex cuts of graphs, which are used to characterize the *l*_{1}-graphs. We have proven that the collection of convex cuts of the gate-sum graph ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ can be expanded by those of *G*_{1} and *G*_{2}. We have then proven the main theorem. For the sake of brevity, we obtained the main result by omitting the proofs of certain lemmas. In section 3, we have presented detailed proofs of those lemmas that were not proved in section 2. Finally, we have presented our conclusions to this study in section 4.

## 2. Convex Cuts and Main Results

Deza and Tuma [12] introduced the concept of convex cuts, which can be used to characterize *l*_{1}-graphs. A *cut* {*A, B*} of *G* is a partition of *V*(*G*) into two nonempty parts. If both *A* and *B* are convex sets, then the cut {*A, B*} is a *convex cut*. A cut {*A, B*} of *G* *cuts* an edge *uv* if *u* ∈ *A* and *v* ∈ *B*. An *edge cut* of *G* is a subset of *E*(*G*) of the form $\left[S,\overline{S}\right]$, where *S* is a nonempty proper subset of *V*(*G*), $\overline{S}=V\backslash S$, and $\left[S,\overline{S}\right]$ is the set of edges with one end in *S* and the other in $\overline{S}$. Similarly, we say that a cut {*A, B*} of *G* *cuts* a subgraph *H* if [*A*∩*V*(*H*), *B*∩*V*(*H*)] is an edge cut of *H*.

Deza and Tuma [12] and Deza et al. [15] proved the following theorem.

**Theorem 2.1**. ([12, 15]) *A graph G is scale-λ-embeddable into a hypercube if and only if there exists a collection ${C}(G)$ of (not necessarily distinct) convex cuts of G such that every edge of G is cut by exactly λ cuts from ${C}(G)$*.

For example, in the graph *K*_{4} in Figure 4, the cuts {{*a*}, {*b, c, d*}}, {{*b*}, {*a, c, d*}}, {{*c*}, {*a, b*, *d*}}, {{*d*}, {*a*, *b, c*}} are convex cuts. Every edge of *K*_{4} is cut by exactly 2 cuts of {{*a*}, {*b, c, d*}}, {{*b*}, {*a, c, d*}}, {{*c*}, {*a, b, d*}}, and {{*d*}, {*a, b, c*}}. By Theorem 2.1, the graph *K*_{4} is scale-2-embeddable into the hypercube *Q*_{4}.

Furthermore, Wang and Zhang [14] showed that the scale of an *l*_{1}-graph can be proportionally amplified.

**Lemma 2.2**. ([14]) *If a graph G is scale*-λ-

*embeddable into a hypercube, then, for any positive integer*

*r*,

*G*

*is scale*-

*r*λ-

*embeddable into a hypercube*.

Let *G*_{1} and *G*_{2} be two *l*_{1}-graphs and ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ be a gate-sum graph of *G*_{1} and *G*_{2}. Without loss of generality, suppose that *G*_{1} is scale-λ-embeddable into some hypercube and *G*_{2} is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections ${C}({G}_{1})$ and ${C}({G}_{2})$ such that every edge of *G*_{1} and *G*_{2} is cut by exactly λ and η cuts, respectively. According to Theorem 2.1 and Lemma 2.2, to prove ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ is an *l*_{1}-graph, it is sufficient to construct a collection ${C}({G}_{1}{\cup}_{H}^{g}{G}_{2})$ of convex cuts of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ such that every edge of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ is cut by exactly the same number of cuts. Now, we construct a collection of convex cuts of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ from the convex cuts of ${C}({G}_{1})$ and ${C}({G}_{2})$.

We now define the *expansion* of convex cuts. Suppose that *H* is a subgraph of *G* and {*A, B*} is a convex cut of *H*. If *G* has a convex cut {*A*′, *B*′} such that *A* ⊆ *A*′ and *B* ⊆ *B*′, then we say that the convex cut {*A, B*} of *H* *expands* the convex cut {*A*′, *B*′} of *G*. We say that the collection ${C}(H)$ *expands* a collection ${C}(G)$ if every convex cut of ${C}(H)$ can expand a convex cut of *G*. We also say that the collection ${C}(H)$ is the *restriction* of ${C}(G)$ on the subgraph *H*.

To enhance the readability of this paper, we list the following three lemmas without proofs. Their proofs have been given in section 3.

**Lemma 2.3**. *Suppose that* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ *is a gate-sum graph of two* *l*_{1}-*graphs* *G*_{1} *and* *G*_{2}. *Then, a convex cut of* *G*_{1} *(or* *G*_{2}) *not cutting* *H* *can expand a convex cut of* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

Next, we will prove that two convex cuts of *G*_{1} and *G*_{2} can expand a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ if they cut the same edges of *H*. Suppose that the convex cut {*A*_{1}, *B*_{1}} of *G*_{1} is cutting *H* and that the cut {*A*_{2}, *B*_{2}} is that of *G*_{2}. Then, {*A*_{1}, *B*_{1}} and {*A*_{2}, *B*_{2}} cut the same edges of *H*. If *A*_{1}∩*A*_{2} ≠ ∅, then *A*_{1}∩*B*_{2} = ∅. If not, *A*_{1}∩*A*_{2} ≠ ∅ and *A*_{1}∩*B*_{2} ≠ ∅, which contradicts the assertion that {*A*_{1}, *B*_{1}} and {*A*_{2}, *B*_{2}} cut the same edges of *H*. Similarly, we have *B*_{1}∩*B*_{2} ≠ ∅ and *B*_{1}∩*A*_{2} = ∅. Because *A*_{i}∪*B*_{i} = *V*(*G*_{i}) (*i* = 1, 2) and *V*(*G*_{1})∩*V*(*G*_{2}) = *V*(*H*), we know that *A*_{1}∩*V*(*H*) = *A*_{1}∩(*A*_{1}∪*B*_{1})∩(*A*_{2}∪*B*_{2}) = *A*_{1}∩*A*_{2} and *A*_{2}∩*V*(*H*) = *A*_{1}∩*A*_{2}. Similarly, *B*_{1}∩*V*(*H*) = *B*_{1}∩*B*_{2} = *B*_{2}∩*V*(*H*). Furthermore, we have that *V*(*H*) = *V*(*G*_{1})∩*V*(*G*_{2}) = (*A*_{1}∪*B*_{1})∩(*A*_{2}∪*B*_{2}) = [*A*_{1}∩(*A*_{2}∪*B*_{2})]∪[*B*_{1}∩(*A*_{2}∪*B*_{2})] = [*A*_{1}∩*A*_{2}]∪[*B*_{1}∩*B*_{2}]. We denote *V*(*H*_{A}) = *A*_{1}∩*A*_{2} and *V*(*H*_{B}) = *B*_{1}∩*B*_{2}. Then, *V*(*H*_{A})∪*V*(*H*_{B}) = *V*(*H*), and we have the following lemma.

**Lemma 2.4**. *Suppose that* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ *is a gate-sum graph of two l_{1}-graphs G_{1} and G_{2}. Assume that {A_{1}, B_{1}} is a convex cut of G_{1} and {A_{2}, B_{2}} is that of G_{2}. If H is l_{1}-rigid, {A_{1}, B_{1}} and {A_{2}, B_{2}} cut the same edges of H. Then, {A_{1}, B_{1}} and {A_{2}, B_{2}} can together expand a convex cut* {

*A*

_{1}∪

_{V(HA)}

*A*

_{2},

*B*

_{1}∪

_{V(HB)}

*B*

_{2}}

*of*${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

If *H* is not *l*_{1}-rigid, then it has more than one kind of collection of convex cuts. Any two collections ${C}({G}_{1})$ and ${C}({G}_{2})$ may not be equal on *H*. Therefore, the convex cuts of ${C}({G}_{1})$ and ${C}({G}_{2})$ may not cut the same edges of *H*.

To solve this problem, we have proven that any kind of collection of convex cuts of *H* can expand two new collections of convex cuts of *G*_{1} and *G*_{2}, respectively, such that they are equal on *H*.

**Lemma 2.5**. *Let H be an isometric subgraph of an l_{1}-graph G. If H is not l_{1}-rigid, each collection ${C}(H)$ of H can expand a collection ${C}(G)$ of G*.

We will now prove the main theorem of this work.

**Theorem 2.6**. *Suppose that* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ *is a gate-sum graph of* *G*_{1} *and* *G*_{2}. *If* *G*_{1} *and* *G*_{2} *are* *l*_{1}-*embeddable, then* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ *is also* *l*_{1}-*embeddable*.

*Proof*: Without loss of generality, suppose that *H* is a gate subgraph of *G*_{1}. Because a gate subgraph is a convex subgraph, *H* is a convex subgraph of *G*_{1}. Then, *H* is an *l*_{1}-graph. Suppose that *G*_{1} is scale-λ-embeddable into some hypercube and *G*_{2} is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections ${C}({G}_{1})$ and ${C}({G}_{2})$ such that every edge of *G*_{1} and *G*_{2} is cut by exactly λ and η cuts, respectively.

If *H* is *l*_{1}-rigid, *H* has only one kind of collection of convex cuts. Then, ${C}({G}_{1})$ and ${C}({G}_{2})$ have the same restriction on *H* (which means that λ = η).

If *H* is not *l*_{1}-rigid, the restriction on *H* of ${C}({G}_{1})$ is not equal to that of ${C}({G}_{2})$. Suppose that λ ≠ η. By Lemma 2.2, *G*_{2} is scale-λη-embeddable into some hypercube. Then, *G*_{2} has a collection ${{C}}^{\prime}({G}_{2})$ such that every edge of *G*_{2} is cut by exactly λη cuts. By Lemma 2.5, every ${C}(H)$ can expand a collection ${C}({G}_{1})$. Obviously, the restriction on *H* of ${{C}}^{\prime}({G}_{2})$ is a kind of ${C}(H)$. Thus, it can expand a new collection ${{C}}^{\prime}({G}_{1})$ of *G*_{1} such that every edge of *G*_{1} is cut by exactly λη cuts.

Hence, there always are two collections ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ for which the restrictions of them on *H* are equal, and every edge of *G*_{1} and *G*_{2} is cut by exactly λη cuts.

As ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ are equal on *H*, there are the same number of convex cuts of ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ cutting *H*. Denote the convex cuts of ${{C}}^{\prime}({G}_{1})$ that are cutting *H* as {*A*_{1}, *B*_{1}}, ..., {*A*_{h}, *B*_{h}} and those of ${{C}}^{\prime}({G}_{2})$ as $\left\{{A}_{1}^{\prime},{B}_{1}^{\prime}\right\},...,\left\{{A}_{h}^{\prime},{B}_{h}^{\prime}\right\}$. Because the restrictions on *H* of ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ are equal, each convex cut of {*A*_{1}, *B*_{1}}, ..., {*A*_{h}, *B*_{h}} must equal one of $\left\{{A}_{1}^{\prime},{B}_{1}^{\prime}\right\},...,\left\{{A}_{h}^{\prime},{B}_{h}^{\prime}\right\}$ on *H*. Without loss of generality, we assume that each pair of {*A*_{i}, *B*_{i}} and $\left\{{A}_{i}^{\prime},{B}_{i}^{\prime}\right\}$ cut the same edges of *H* (1 ≤ *i* ≤ *h*). By Lemma 2.4, each pair of convex cuts {*A*_{i}, *B*_{i}} and $\left\{{A}_{i}^{\prime},{B}_{i}^{\prime}\right\}$ can together expand a convex cut $\left\{{A}_{i}\cup {A}_{i}^{\prime},{B}_{i}\cup {B}_{i}^{\prime}\right\}$ of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ (1 ≤ *i* ≤ *h*). Then, every edge of *H* is cut by $\left\{{A}_{i}\cup {A}_{i}^{\prime},{B}_{i}\cup {B}_{i}^{\prime}\right\}$ to give exactly λη cuts (1 ≤ *i* ≤ *h*).

By Lemma 2.3, the convex cuts of ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ that do not cut *H* can expand the convex cuts of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ that do not cut *H*.

Now, the convex cuts $\left\{{A}_{i}\cup {A}_{i}^{\prime},{B}_{i}\cup {B}_{i}^{\prime}\right\}$ for 1 ≤ *i* ≤ *h*, together with the convex cuts of ${{C}}^{\prime}({G}_{1})$ and ${{C}}^{\prime}({G}_{2})$ that do not cut *H*, form a collection of convex cuts of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, such that every edge of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ is cut by λη convex cuts. Therefore, by Theorem 2.1, the graph ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ is scale-λη-embedded into some hypercube. This completes the proof.

Note that, for any graph, a single vertex is a gate subgraph. A *cycle* is a closed path that originates and terminates at the same vertex. A graph is bipartite if and only if it contains no odd cycles [4]. Therefore, for any edge *e* = *uv* of a bipartite graph, there is no vertex *a* such that *d*(*u, a*) = *d*(*v, a*). The subgraph induced by an edge is then a gate subgraph in a bipartite graph. Obviously, both *G*_{1} and *G*_{2} are isometric subgraphs of the graphs *G*_{1}∪_{v}*G*_{2} and *G*_{1}∪_{uv}*G*_{2}. The following corollaries can be immediately obtained from Theorem 2.6.

**Corollary 2.7**. ([11]). Let *G*_{1} and *G*_{2} be two *l*_{1}-graphs. *G*_{1}∪_{v}*G*_{2} is an *l*_{1}-graph.

**Corollary 2.8**. ([14]). Let *G*_{1} and *G*_{2} be two *l*_{1}-graphs. If at least one of them is bipartite, *G*_{1}∪_{uv}*G*_{2} is an *l*_{1}-graph.

## 3. Proofs of Lemmas 2.3–2.5

### 3.1. Proof of Lemma 2.3

First, we need the following lemma.

**Lemma 3.1**. *Suppose that* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ *is a gate-sum graph of* *G*_{1} *and* *G*_{2}. *If* *H* *is a gate subgraph of* *G*_{1}, *then* *G*_{2} *is a convex subgraph of* ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

*Proof*: If *G*_{2} is not a convex subgraph of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, there are two vertices *x*_{1} and *x*_{2} lying in *G*_{2} such that the shortest path *P*_{x1x2} passes through a vertex *v*_{3} of *G*_{1}. As this shortest path must pass through the vertices of the gate subgraph *H* of *G*_{1}, there are two vertices ${x}_{1}^{\prime}$ and ${x}_{2}^{\prime}$ of *H* on the *x*_{1}, *v*_{3}-path and *x*_{2}, *v*_{3}-path of *P*_{x1x2}, respectively. Note that both *G*_{1} and *G*_{2} are isometric subgraphs of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. It is clear that the segment from ${x}_{1}^{\prime}$ to ${x}_{2}^{\prime}$ of *P*_{x1x2} is a shortest path ${P}_{{x}_{1}^{\prime}{x}_{2}^{\prime}}$. Then, we have ${P}_{{x}_{1}^{\prime}{x}_{2}^{\prime}}={P}_{{x}_{1}^{\prime}{v}_{3}}+{P}_{{v}_{3}{x}_{2}^{\prime}}$.

As *H* is a gate subgraph of *G*_{1}, there exists a unique gate *a*_{3} of *v*_{3} in *H* such that ${P}_{{x}_{1}^{\prime}{v}_{3}}={P}_{{x}_{1}^{\prime}{a}_{3}}+{P}_{{a}_{3}{v}_{3}}$ and ${P}_{{x}_{2}^{\prime}{v}_{3}}={P}_{{x}_{2}^{\prime}{a}_{3}}+{P}_{{a}_{3}{v}_{3}}$. Then, we have that ${P}_{{x}_{1}^{\prime}{x}_{2}^{\prime}}={P}_{{x}_{1}^{\prime}{v}_{3}}+{P}_{{v}_{3}{x}_{2}^{\prime}}={P}_{{x}_{1}^{\prime}{a}_{3}}+{P}_{{a}_{3}{v}_{3}}+{P}_{{x}_{2}^{\prime}{a}_{3}}+{P}_{{a}_{3}{v}_{3}}>{P}_{{x}_{1}^{\prime}{a}_{3}}+{P}_{{a}_{3}{x}_{2}^{\prime}}$, which contradicts the assertion that ${P}_{{x}_{1}^{\prime}{x}_{2}^{\prime}}$ is a shortest path. Thus, *G*_{2} is a convex subgraph of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

**Proof of Lemma 2.3**. Without loss of generality, suppose that *H* is a gate subgraph of *G*_{1}. We need only prove that a convex cut of *G*_{1} or *G*_{2} that does not cut *H* can expand a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

**Case 1**. A convex cut of *G*_{1} that does not cut *H* can expand that of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

Suppose {*A, B*} is a convex cut of *G*_{1} that does not cut *H*. Without loss of generality, we assume that *V*(*H*) ⊆ *B*. We now prove that {*A, B*∪_{V(H)}*V*(*G*_{2})} is a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ that does not cut *H* and is expanded by {*A, B*}.

If *A* is not a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, there are two vertices *v*_{1} and *v*_{2} belonging to *A* such that *P*_{v1v2} of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ passes through a vertex *v*_{3} of *G*[*B*]∪_{H}*G*_{2}. Therefore, *P*_{v1v2} = *P*_{v1v3} + *P*_{v3v2}. If all vertices of *P*_{v1v2} lie entirely in *G*_{1}, *A* cannot be a convex set of *G*_{1}. Without loss of generality, suppose that *v*_{3} lies in *G*_{2}. Note that *H* is a gate subgraph of *G*_{1}. There are two gates *x*_{1} of *v*_{1} and *x*_{2} of *v*_{2} in *H*, and these two gates lie in *P*_{v1v3} and *P*_{v3v2}, respectively. Then, we have that *P*_{v1v2} = *P*_{v1x1} + *P*_{x1v3} + *P*_{v3x2} + *P*_{x2v2}. As *G*_{1} is an isometric subgraph of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, there is some *P*_{x1x2} that lies entirely in *G*_{1}, and its length equals that of *P*_{x1x2} of *G*_{2}. Then, *P*_{v1v2} = *P*_{v1x1} + *P*_{x1x2} + *P*_{x2v2}, and *P*_{v1v2} lies entirely in *G*_{1}. As *P*_{v1v2} passes through the vertices of *H*, and *H* belongs to *B*, *A* cannot be a convex set of *G*_{1}. Therefore, *A* is a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

If *B*∪_{V(H)}*V*(*G*_{2}) is not a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, there are two vertices *v*_{4} and *v*_{5} belonging to *B*∪_{V(H)}*V*(*G*_{2}) such that *P*_{v4v5} passes through a vertex *v*_{6} in *A* and *P*_{v4v5} = *P*_{v4v6} + *P*_{v6v5}. Obviously, the path *P*_{v4v6} does not intersect with *P*_{v6v5} at any internal vertices. The segment of a shortest path is still a shortest path. This means that *v*_{6} has two internally disjoint paths *P*_{v6v4} and *P*_{v6v5} that connect with the vertices in *H*. Thus, *v*_{6} has at least two gates, which contradicts the statement that the gate is unique.

Both *A* and *B*∪_{V(H)}*V*(*G*_{2}) are convex sets of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, and they contain all vertices of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Thus, {*A, B*∪_{V(H)}*V*(*G*_{2})} is a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Furthermore, note that *A* = *A* and *B* ⊆ *B* ∪ _{V(H)}*V*(*G*_{2}), and so the convex cut {*A, B* ∪ _{V(H)}*V*(*G*_{2})} is expanded by the convex cut {*A, B*} of *G*_{1}.

**Case 2**. A convex cut of *G*_{2} that does not cut *H* can expand that of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

Suppose that {*C, D*} is a convex cut of *G*_{2} that does not cut *H*. Without loss of generality, we assume that *H* ⊆ *D*. We now prove that the cut {*C, D* ∪ _{V(H)}*V*(*G*_{1})} is a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ and is expanded by {*C, D*}.

By Lemma 3.1, it is obvious that *C* is a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ because *C* is a convex set of *G*_{2} and *G*_{2} a convex subgraph of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

Suppose that the vertex set *D* ∪ _{V(H)}*V*(*G*_{1}) is not a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. There will be two vertices *v*_{1}, *v*_{2} of *D* ∪ _{V(H)}*V*(*G*_{1}) such that the shortest path *P*_{v1v2} passes through a vertex *v*_{3} of *C*. Let *P*_{v1v3} and *P*_{v3v2} denote the two segments of *P*_{v1v2} divided by *v*_{3}. Because the vertices *v*_{1}, *v*_{2} belong to *D* ∪ _{V(H)}*V*(*G*_{1}), we can find two vertices ${v}_{1}^{\prime},{v}_{2}^{\prime}$ of *D* such that ${v}_{1}^{\prime}\in {P}_{{v}_{1}{v}_{3}}$ and ${v}_{2}^{\prime}\in {P}_{{v}_{3}{v}_{2}}$. Note that both *G*_{1} and *G*_{2} are isometric subgraphs of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. It is clear that the segment from ${v}_{1}^{\prime}$ to ${v}_{2}^{\prime}$ of the path *P*_{v1v2} is a shortest path, and it passes through the vertex *v*_{3} of *C*, which contradicts the assertion that *D* is a convex set of *G*_{2}. Therefore, *D* ∪ _{V(H)}*V*(*G*_{1}) is a convex set of *G*.

As *C* and *D* ∪ _{V(H)}*V*(*G*_{1}) are convex sets of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, {*C, D* ∪ _{V(H)}*V*(*G*_{1})} is a convex cut of *G* and its two convex sets contain *C* and *D*, respectively. It follows that the convex cut {*C, D*} of *G*_{2} that does not cut *H* expands the convex cut {*C, D* ∪ _{V(H)}*V*(*G*_{1})} of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. This completes the proof.

□

### 3.2. Proof of Lemma 2.4

*Proof*: Let *G*_{1} and *G*_{2} be two *l*_{1}-graphs and ${G}_{1}{\cup}_{H}^{g}{G}_{2}$ be the gate-sum graph of *G*_{1} and *G*_{2}. By Theorem 2.1, there are two collections ${C}({G}_{1})$ and ${C}({G}_{2})$ such that every edge of *G*_{1} and *G*_{2} is cut by exactly λ and η cuts, respectively, as *H* is *l*_{1}-rigid, ${C}({G}_{1})$ and ${C}({G}_{2})$ must be equal on *H*. For any convex cut {*A*_{1}, *B*_{1}} of ${C}({G}_{1})$, we can find a convex cut {*A*_{2}, *B*_{2}} of ${C}({G}_{2})$ that cuts the same edge of *H*.

Without loss of generality, suppose that *H* is a gate subgraph of an *l*_{1}-graph *G*_{1}. Suppose that *x*_{1} of *V*(*H*) is the gate of *v*_{1} in *G*_{1}. If *v*_{1} and *x*_{1} belong to different convex sets, assume that *v*_{1} lies in *A*_{1} and *x*_{1} belongs to *B*_{1}∩*V*(*H*). There will be a vertex *u* in *A*_{1}∩*V*(*H*) such that the shortest path *P*_{v1u} must pass through the vertices of *B*_{1}, which contradicts the assertion that *A*_{1} is a convex set. Then, both *v*_{1} and *x*_{1} belong to the same convex set *A*_{1} or *B*_{1}.

Without loss of generality, suppose that *v*_{1} and *x*_{1} belong to *A*_{1}. We now show that {*A*_{1} ∪_{V(HA)}*A*_{2}, *B*_{1} ∪ _{V(HB)}*B*_{2}} is a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. First, we prove that *A*_{1} ∪ _{V(HA)}*A*_{2} is a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Consider two vertices *v*_{1} and *v*_{2} that belong to *A*_{1} ∪ _{V(HA)}*A*_{2}.

**Case 1**. Both *v*_{1} and *v*_{2} lie in *A*_{2}.

As *A*_{2} is a convex subset of *G*_{2} and *G*_{2} is a convex subgraph of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, *A*_{2} is a convex subset of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Obviously, *P*_{v1v2} lies entirely in *A*_{2}.

**Case 2**. The vertex *v*_{1} lies in *A*_{1} and *v*_{2} lies in *A*_{2}.

Because *v*_{1} lies in *A*_{1} and *v*_{2} lies in *A*_{2}, the gate *x*_{1} of *v*_{1} belongs to *A*_{1}∩*V*(*H*). As {*A*_{1}, *B*_{1}} and {*A*_{2}, *B*_{2}} cut the same edges of *H*, we have that *A*_{1}∩*V*(*H*) = *A*_{2}∩*V*(*H*) and *x*_{1} also belongs to *A*_{2}. Therefore, the shortest path *P*_{v1v2} must pass through the vertices of *H*.

If *P*_{v1v2} passes through the gate *x*_{1} of *v*_{1}, we have that *P*_{v1v2} = *P*_{v1x1} + *P*_{x1v2}. Note that both *G*_{1} and *G*_{2} are isometric subgraphs of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. As both *v*_{1} and *x*_{1} belong to *A*_{1} and *A*_{1} is a convex set, the path *P*_{v1x1} lies entirely in *A*_{1}. Similarly, *v*_{2} and *x*_{1} belong to *A*_{2}, which is a convex set. Hence, *P*_{v2x1} lies entirely in *A*_{2}. Thus, the shortest path *P*_{v1v2} lies entirely in *A*_{1} ∪ _{V(HA)}*A*_{2}.

If there is a shortest path *P*_{v1v2} that does not pass through the gate *x*_{1} of *v*_{1}, *P*_{v1v2} will pass through a vertex *x*_{3} of *V*(*H*), which is not the gate of *v*_{1}, and *P*_{v1v2} = *P*_{v1x3} + *P*_{x3v2}.

We now prove that *x*_{3} belongs to *A*_{1}∩*V*(*H*). If this is not the case, then *x*_{3} lies in *B*_{1}∩*V*(*H*), and so *P*_{v1x3} = *P*_{v1x1} + *P*_{x1x3} and *P*_{x1v2} < *P*_{x1x3} + *P*_{x3v2}. Furthermore, *P*_{v1x3} + *P*_{x3v2} = *P*_{v1x1} + *P*_{x1x3} + *P*_{x3v2} > *P*_{v1x1} + *P*_{x1v2}, which contradicts the assertion that *P*_{v1v2} passes through *x*_{3}, but does not pass through the gate *x*_{1}.

As *v*_{1} and *x*_{3} belong to *A*_{1}, and *x*_{3} and *v*_{2} belong to *A*_{2}, we have that *P*_{v1x3} lies entirely in *A*_{1} and *P*_{x3v2} lies entirely in *A*_{2}. Therefore, *P*_{v1v2} = *P*_{v1x3} + *P*_{x3v2} lies entirely in *A*_{1} ∪ _{V(HA)}*A*_{2}.

Hence, for any vertex *v*_{1} of *A*_{1} and any vertex *v*_{2} of *A*_{2}, *P*_{v1v2} lies entirely in *A*_{1} ∪ _{V(HA)}*A*_{2}. This proves case 2.

**Case 3**. Both *v*_{1} and *v*_{2} lie in *A*_{1}.

If *P*_{v1v2} does not pass through the vertices of *G*_{2}, then *P*_{v1v2} lies in *G*_{1}. Note that *A*_{1} is a convex subgraph of *G*_{1}, and *P*_{v1v2} lies in *A*_{1}. If *P*_{v1v2} passes through the vertices of *G*_{2}, it must pass through a vertex *v*_{3} of *A*_{2}. From case 2, we know that both *P*_{v1v3} and *P*_{v3v2} lie in *A*_{1} ∪ _{V(HA)}*A*_{2} and that *P*_{v1v2} lies entirely in *A*_{1} ∪ _{V(HA)}*A*_{2}.

Summarizing the above three cases, for any two vertices *v*_{1} and *v*_{2} of *A*_{1} ∪ _{V(HA)}*A*_{2}, we have that the shortest path *P*_{v1v2} lies entirely in *A*_{1} ∪ _{V(HA)}*A*_{2}. It follows that *A*_{1} ∪ _{V(HA)}*A*_{2} is a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

A similar proof shows that the set *B*_{1} ∪ _{V(HB)}*B*_{2} is also a convex set of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$. Then, {*A*_{1} ∪ _{V(HA)}*A*_{2}, *B*_{1} ∪ _{V(HB)}*B*_{2}} is a convex cut of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$, and its two convex sets contain vertex sets *A*_{1}, *A*_{2} and *B*_{1}, *B*_{2}, respectively. Thus, {*A*_{1}, *B*_{1}} of *G*_{1} and {*A*_{2}, *B*_{2}} of *G*_{2} together expand the convex cut {*A*_{1} ∪ _{V(HA)}*A*_{2}, *B*_{1} ∪ _{V(HB)}*B*_{2}} of ${G}_{1}{\cup}_{H}^{g}{G}_{2}$.

### 3.3. Proof of Lemma 2.5

To study the expansion of the collection of convex cuts, we have introduced a new characteristic of *l*_{1}-graphs. Shpectorov [9] and Deza and Grishukhin [10] characterized *l*_{1}-graphs as follows:

**Theorem 3.2**. ([9, 10]) *A graph G is an l_{1}-graph if and only if it is an isometric subgraph of the Cartesian product of cocktail party graphs and half-cubes*.

Suppose that *H* is an isometric subgraph of an *l*_{1}-graph *G*; *H* is also an *l*_{1}-graph. By Theorem 3.2, *H* is an isometric subgraph of the Cartesian product of some cocktail party graphs and half-cubes, and *G* is that of larger cocktail party graphs and larger half-cubes. To expand the collection of convex cuts of *H* to *G*, we need only expand the collection of convex cuts of the cocktail party graph and half-cube to a larger cocktail party graph and a larger half-cube, respectively. As the half-cube is *l*_{1}-rigid, it has a unique collection of convex cuts. Note that $\frac{1}{2}{Q}_{m}$ is a subgraph of $\frac{1}{2}{Q}_{n}$. Thus, we have that any collection ${C}(\frac{1}{2}{Q}_{m})$ of $\frac{1}{2}{Q}_{m}$ can expand a collection ${C}(\frac{1}{2}{Q}_{n})$ of $\frac{1}{2}{Q}_{n}$ (*m* ≤ *n*). We need only examine whether any collection ${C}({K}_{m\times 2})$ can expand a collection ${C}({K}_{n\times 2})$ (*m* ≤ *n*).

We require the definition of a vertex-transitive graph. An *automorphism* of a (simple) graph *G* is a permutation π of *V*(*G*) that has the property that (*u, v*) is an edge of *G* if and only if (π(*u*), π(*v*)) is an edge of *G*. The set of all automorphisms of *G*, with the composition operation, is a group. This group is called the *automorphism group of* *G*. A graph *G* is *vertex-transitive* if the automorphism group of *G* acts transitively on *V*(*G*) [16, 17].

In other words, a *vertex-transitive graph* is a graph *G* such that, given any two vertices *v*_{1} and *v*_{2} of *G*, there is some automorphism *f*:*V*(*G*) → *V*(*G*) such that *f*(*v*_{1}) = *v*_{2}.

For a complete graph *K*_{n}, we constructed its collection of convex cuts. Without loss of generality, assume that *V*(*K*_{n}) = {*v*_{1}, ..., *v*_{n}}. From Theorem 3.2, *K*_{n} is an *l*_{1}-graph. Suppose that *K*_{n} is scale-λ-embeddable into a hypercube. Theorem 2.1 implies that there is a collection ${C}({K}_{n})$ such that every edge *uv* is cut by λ cuts (*u, v* belong to *K*_{n} and λ is even). We assume that {*S*_{1}, *V*(*K*_{n}) − *S*_{1}} is a convex cut of ${C}({K}_{n})$, and that both *S*_{1} and *V*(*K*_{n}) − *S*_{1} are convex sets of *V*(*K*_{n}) (|*S*_{1}| = *q*). As the complete graph is vertex-transitive, each *S*_{i} constructs a convex cut of *K*_{n} of the form *S*_{i} ⊆ *V*(*K*_{n}), |*S*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$). Then, we have that all convex cuts {*S*_{i}, *V*(*K*_{n}) − *S*_{i}}, |*S*_{i}| = *q* $(1\le i\le (\frac{n}{q}))$, form a collection of convex cuts of *K*_{n} such that every edge of *K*_{n} is cut by the same cuts.

Obviously, there are $\left(\frac{n}{q}\right)$ different convex cuts, and each convex cut acts on *q*(*n* − *q*) edges. Note that the complete graph *K*_{n} has $\frac{n(n-1)}{2}$ edges and is vertex-transitive. Thus, we have that $\lambda =\frac{\left(\frac{n}{q}\right)q(n-q)}{\frac{n(n-1)}{2}}=2\left(\frac{n-2}{q-1}\right)$.

For *m* ≤ *n*, we can now prove that the collection ${C}({K}_{m\times 2})$ of *K*_{m × 2} can expand a collection ${C}({K}_{n\times 2})$ of *K*_{n × 2}.

**Theorem 3.3**. *Let* *K*_{n × 2} *be a cocktail party graph and* *K*_{m × 2} *be a cocktail party subgraph of* *K*_{n × 2}. *Every collection* ${C}({K}_{m\times 2})$ *of* *K*_{m × 2} *can expand a collection* ${C}({K}_{n\times 2})$ *of* *K*_{n × 2}.

*Proof*: Obviously, the cocktail party graph *K*_{n × 2} has a complete subgraph *K*_{n}. Without loss of generality, assume that *V*(*K*_{n}) = {*v*_{1}, ..., *v*_{n}}, $V({K}_{n}^{\prime})=\left\{{v}_{1}^{\prime},...,{v}_{n}^{\prime}\right\}$, and $V({K}_{n\times 2})=\left\{{v}_{1},...,{v}_{n},{v}_{1}^{\prime},...,{v}_{n}^{\prime}\right\}$ such that ${d}_{{K}_{n\times 2}}({v}_{j},{v}_{j}^{\prime})=2$ (1 ≤ *j* ≤ *n*), ${d}_{{K}_{n\times 2}}({v}_{i},{v}_{j})={d}_{{K}_{n\times 2}}({v}_{i},{v}_{j}^{\prime})=1$ (*i* ≠ *j*). If the vertex set *S* is a subset of *V*(*K*_{n × 2}), then the vertex set *S*′ = {*x*′|*x* ∈ *S*} is a subset of $V({K}_{n\times 2}^{\prime})$.

First, we prove that every convex cut of *K*_{n × 2} has only two forms: $\left\{S\cup (V({K}_{n}^{\prime})-{S}^{\prime}),{S}^{\prime}\cup (V({K}_{n})-S)\right\}$ and $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$.

Suppose that {*A, B*} is a convex cut of *K*_{n × 2}. If *x* belongs to *A*, *x*′ will belong to *B*. If not, both *x* and *x*′ belong to *A*, and *A* is a convex subset of *V*(*K*_{n × 2}); all vertices of *V*(*K*_{n × 2}) will then belong to *A*. Furthermore, *B* is an empty set, which contradicts both *A* and *B* being nonempty. We now have that the vertex sets *S* and *S*′ belong to different convex sets of {*A, B*}. Without loss of generality, suppose that *S* ⊆ *A* and *S*′ ⊆ *B*. If *V*(*K*_{n}) − *S* ⊆ *A*, then $V({K}_{n}^{\prime})-{S}^{\prime}\subseteq B$ and $\left\{A,B\right\}=\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$. If *V*(*K*_{n}) − *S* ⊆ *B*, then $V({K}_{n}^{\prime})-{S}^{\prime}\subseteq A$ and $\left\{A,B\right\}=\{S\cup (V({K}_{n}^{\prime})-{S}^{\prime}$), ${S}^{\prime}\cup (V({K}_{n})-S)\}$.

Thus, the convex cut of *K*_{n × 2} has only two forms, $\left\{S\cup (V({K}_{n}^{\prime})-{S}^{\prime}),{S}^{\prime}\cup (V({K}_{n})-S)\right\}$ and $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$.

Second, we prove that the collection of convex cuts $\left\{{S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}),{S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i})\right\}$, |*S*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$), together with some $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ make the cocktail graph *K*_{n × 2} embeddable into some cubes.

For every edge *uv* in *K*_{n}, *uv* is cut by the convex cut $\left\{{S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}),{S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i})\right\}$. We have that $u\in ({S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}))\cap V({K}_{n})={S}_{i}$ and $v\in ({S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i}))\cap V({K}_{n})=V({K}_{n})-{S}_{i}$, or *u* ∈ *V*(*K*_{n}) − *S*_{i} and *v* ∈ *S*_{i}. Note that |*S*_{i}| = *q* and *V*(*K*_{n}) has *n* vertices, so the number of convex cuts that cut edge *uv* is $2\left(\frac{n-2}{q-1}\right)$. This is similar to each edge *u*′*v*′ of ${K}_{n}^{\prime}$.

If *u* ∈ *K*_{n} and ${v}^{\prime}\in {K}_{n}^{\prime}$, *uv*′ is cut by the convex cut $\left\{{S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}),{S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i})\right\}$. We have that $u\in ({S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}))\cap V({K}_{n})={S}_{i}$, ${v}^{\prime}\in ({S}_{i}^{\prime}\cup \left\{V({K}_{n})-{S}_{i}\right\})\cap V({K}_{n}^{\prime})={S}_{i}^{\prime}$, ${u}^{\prime}\in {S}_{i}^{\prime}$, and *v* ∈ *S*_{i}, or $u\in ({S}_{i}^{\prime}\cup \left\{V({K}_{n})-{S}_{i}\right\})\cap V({K}_{n})=V({K}_{n})-{S}_{i}$, ${v}^{\prime}\in ({S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}))\cap V({K}_{n}^{\prime})=V({K}_{n}^{\prime})-{S}_{i}^{\prime}$, ${u}^{\prime}\in V({K}_{n}^{\prime})-{S}_{i}^{\prime}$, and *v* ∈ *V*(*K*_{n}) − *S*_{i}. Note that $\left|{S}_{i}\right|=\left|{S}_{i}^{\prime}\right|=q$ and $|V({K}_{n})-{S}_{i}|=|V({K}_{n}^{\prime})-{S}_{i}^{\prime}|=n-q$, so the number of convex cuts that cut edge *uv*′ is $\left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{n-q-2}\right)=\left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)$.

As $\frac{n-\sqrt{n}}{2}\le q\le \frac{n+\sqrt{n}}{2}$, $2\left(\frac{n-2}{q-1}\right)\ge \left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)$. If $2\left(\frac{n-2}{q-1}\right)=\left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)$, every edge of *K*_{n × 2} is cut by $2\left(\frac{n-2}{q-1}\right)$ cuts.

If $2\left(\frac{n-2}{q-1}\right)>\left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)$, then $\frac{n-\sqrt{n}}{2}<q<\frac{n+\sqrt{n}}{2}$. Obviously, $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ is a convex cut of *K*_{n × 2}, which only cuts the edges with one end vertex in *K*_{n} and the other one in ${K}_{n}^{\prime}$. Then, the collection $\left\{{S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}),{S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i})\right\}$ ($(1\le i\le \left(\frac{n}{q}\right))$) together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ form a new collection ${{C}}^{\prime}({K}_{n\times 2})$ such that every edge of *K*_{n × 2} is cut by $2\left(\frac{n-2}{q-1}\right)$ cuts.

Let $2\left(\frac{n-2}{q-1}\right)<\left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)$. If *n* is even, choose *T*_{i} ⊆ *V*(*K*_{n}) such that $\left|{T}_{i}\right|=\frac{n}{2}$ ($(1\le i\le \left(\frac{n}{\frac{n}{2}}\right))$. If *n* is odd, choose *T*_{i} ⊆ *V*(*K*_{n}) such that $\left|{T}_{i}\right|=\frac{n+1}{2}$ $(1\le i\le (\frac{n}{\frac{n+1}{2}}))$. Then, $\left\{{T}_{i}\cup (V({K}_{n}^{\prime})-{T}_{i}^{\prime}),{T}_{i}^{\prime}\cup (V({K}_{n})-{T}_{i})\right\}$ is a convex cut of *K*_{n × 2}. Obviously, the number of edges with both vertices in *V*(*K*_{n}) (or $V({K}_{n}^{\prime})$) that are cut by $\left\{{T}_{i}\cup (V({K}_{n}^{\prime})-{T}_{i}^{\prime}),{T}_{i}^{\prime}\cup (V({K}_{n})-{T}_{i})\right\}$ is greater than the number of edges that are cut by the same cut with one end vertex in *K*_{n} and the other vertex in ${K}_{n}^{\prime}$. Thus, the collection $\left\{{S}_{i}\cup (V({K}_{n}^{\prime})-{S}_{i}^{\prime}),{S}_{i}^{\prime}\cup (V({K}_{n})-{S}_{i})\right\}$ ($(1\le i\le \left(\frac{n}{q}\right))$) together with $\{{T}_{i}\cup (V({K}_{n}^{\prime})-{T}_{i}^{\prime}),$ ${T}_{i}^{\prime}\cup (V({K}_{n})-{T}_{i})\}$ $(1\le i\le \left(\frac{n}{\frac{n}{2}}\right)$ $(\mathrm{\text{or}}\left(\frac{n}{\frac{n+1}{2}}\right)))$) and $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ form a new collection ${{C}}^{\prime}({K}_{n\times 2})$ such that every edge of *K*_{n × 2} is cut by $2\left(\frac{n-2}{q-1}\right)+2a\left(\frac{n-2}{\frac{n}{2}-1}\right)$ cuts. The constant *a* is the minimal number such that $2\left(\frac{n-2}{q-1}\right)+2a\left(\frac{n-2}{\frac{n}{2}-1}\right)\ge \left(\frac{n-2}{q-2}\right)+\left(\frac{n-2}{q}\right)+a(\left(\frac{n-2}{\frac{n}{2}-2}\right)+(\frac{n-2}{\frac{n}{2}}))$.

Third, we prove that every collection of convex cuts of *K*_{m × 2} can expand that of *K*_{n × 2} (*m* ≤ *n*).

Similarly, each convex cut of *K*_{m × 2} has only two forms: $\left\{A\cup (V({K}_{m}^{\prime})-{A}^{\prime}),{A}^{\prime}\cup (V({K}_{m})-A)\right\}$, and $\left\{V({K}_{m}),V({K}_{m}^{\prime})\right\}$.

Obviously, (*V*(*K*_{m}) − *A*) ⊆ (*V*(*K*_{n}) − *A*) and $(V({K}_{m}^{\prime})-{A}^{\prime})\subseteq (V({K}_{n}^{\prime})-{A}^{\prime})$. Then, each convex cut $\left\{A\cup (V({K}_{m}^{\prime})-{A}^{\prime}),{A}^{\prime}\cup (V({K}_{m})-A)\right\}$ of ${C}({K}_{m\times 2})$ can expand a convex cut $\left\{A\cup (V({K}_{n}^{\prime})-{A}^{\prime}),{A}^{\prime}\cup (V({K}_{n})-A)\right\}$ of ${C}({K}_{n\times 2})$. Similarly, the convex cut $\left\{V({K}_{m}),V({K}_{m}^{\prime})\right\}$ expands the cut $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$.

Assume that |*A*_{i}| = |*A*_{j}| is true for all convex cuts of ${C}({K}_{m\times 2})$ except the convex cut $\left\{V({K}_{m}),V({K}_{m}^{\prime})\right\}$. This means that $\left\{{A}_{i}\cup (V({K}_{m}^{\prime})-{A}_{i}^{\prime}),{A}_{i}^{\prime}\cup (V({K}_{m})-{A}_{i})\right\}$, |*A*_{i}| = *q* ($(1\le i\le \left(\frac{m}{q}\right))$). Then, all of the cuts together with $\left\{V({K}_{m}),V({K}_{m}^{\prime})\right\}$ expand a collection of convex cuts of *K*_{n × 2}, in the form $\left\{{A}_{i}\cup (V({K}_{n}^{\prime})-{A}_{i}^{\prime}),{A}_{i}^{\prime}\cup (V({K}_{n})-{A}_{i})\right\}$, |*A*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$), together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$. By the second part, $\left\{{A}_{i}\cup (V({K}_{n}^{\prime})-{A}_{i}^{\prime}),{A}_{i}^{\prime}\cup (V({K}_{n})-{A}_{i})\right\}$, |*A*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$), together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ ensure that every edge of the graph *K*_{n × 2} is cut by the same cuts.

Let |*A*_{i}| ≠ |*A*_{j}| for some *i* and *j* of the convex cuts of ${C}({K}_{m\times 2})$. Without loss of generality, suppose that ${C}({K}_{m\times 2})$ has three kinds of convex cuts, formed as $\left\{{A}_{i}\cup (V({K}_{n}^{\prime})-{A}_{i}^{\prime}),{A}_{i}^{\prime}\cup (V({K}_{n})-{A}_{i})\right\}$, |*A*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$), and $\left\{{B}_{i}\cup (V({K}_{n}^{\prime})-{B}_{i}^{\prime}),{B}_{i}^{\prime}\cup (V({K}_{n})-{B}_{i})\right\}$, |*B*_{i}| = *p* ($(1\le i\le \left(\frac{n}{p}\right))$), together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$. By the above discussion, all of the convex cuts $\left\{{A}_{i}\cup (V({K}_{n}^{\prime})-{A}_{i}^{\prime}),{A}_{i}^{\prime}\cup (V({K}_{n})-{A}_{i})\right\}$, |*A*_{i}| = *q* ($(1\le i\le \left(\frac{n}{q}\right))$), together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ expand a collection ${{C}}_{1}({K}_{n\times 2})$ of convex cuts of *K*_{n × 2} such that every edge of *K*_{n × 2} is cut by the same cuts. Similarly, all of the convex cuts $\left\{{B}_{i}\cup (V({K}_{n}^{\prime})-{B}_{i}^{\prime}),{B}_{i}^{\prime}\cup (V({K}_{n})-{B}_{i})\right\}$, |*B*_{i}| = *p* ($(1\le i\le \left(\frac{n}{p}\right)$), together with $\left\{V({K}_{n}),V({K}_{n}^{\prime})\right\}$ expand a collection ${{C}}_{2}({K}_{n\times 2})$ of convex cuts of *K*_{n × 2} such that every edge of *K*_{n × 2} is cut by the same cuts.

Obviously, the collection ${{C}}_{1}({K}_{n\times 2})$ together with the collection ${{C}}_{2}({K}_{n\times 2})$ is still a collection of convex cuts of *K*_{n × 2} such that every edge of *K*_{n × 2} is cut by the same cuts.

Therefore, every collection ${C}({K}_{m\times 2})$ of *K*_{m × 2} can expand a collection ${C}({K}_{n\times 2})$ such that every edge of *K*_{n × 2} is cut by the same number of cuts.

We have that, for each cocktail party graph and half-cube, the collection ${C}(\frac{1}{2}{Q}_{m})$ can expand a collection ${C}(\frac{1}{2}{Q}_{n})$, and the collection ${C}({K}_{m\times 2})$ can expand a collection ${C}({K}_{n\times 2})$ (*m* ≤ *n*). By Theorem 3.2, we can prove that the collection of convex cuts of an *l*_{1}-graph can expand that of a larger *l*_{1}-graph.

Hammack et al. [6] introduced the *Cartesian product* *G□H* of two graphs *G* and *H* as the graph whose vertex set is the Cartesian product *V*(*G*) × *V*(*H*). Two vertices (*u, v*) and (*u*′, *v*′) are adjacent in *G□H* if and only if *u* = *u*′ and *v* is adjacent to *v*′ in *H*, or *v* = *v*′ and *u* is adjacent to *u*′ in *G*. Thus,

*V*(*G□H*) = {(*u, v*)|*u* ∈ *V*(*G*) and *v* ∈ *V*(*H*)}

*E*(*G□H*) = {(*u, v*)(*u*′, *v*′)|*u* = *u*′, *vv*′ ∈ *E*(*H*), or *uu*′ ∈

*E*(*G*), *u* = *u*′}

The graphs *G* and *H* are called *factors* of the product *G□H*. Hammack et al. proved the following lemmas.

**Lemma 3.4**. ([6]) *A subgraph* *W* *of* *G* = *G*_{1}□⋯□*G*_{n} *is convex if and only if* *W* = *W*_{1}□⋯□*W*_{n}, *where each* *W*_{i} *is convex in* *G*_{i}.

**Lemma 3.5**. ([6]) *If* *G* = *G*_{1}□⋯□*G*_{n} *and* *x, y* ∈ *V*(*G*), *then*

For any index 1 ≤ *i* ≤ *n*, *p*_{i} is a projection map *p*_{i}:*G*_{1}□⋯□*G*_{n} → *G*_{i}, defined as *p*_{i}(*x*_{1}, *x*_{2}, ..., *x*_{n}) = *x*_{i}.

We can now prove that the convex cut of a Cartesian product can be represented by the convex cuts of all factors.

**Theorem 3.6**. *The cut* {*A, B*} *is a convex cut of a graph* *G* = *G*_{1}□⋯□*G*_{n} *if and only if* {*A, B*} *has the form* {*V*(*G*_{1}) × ⋯ × *V*(*G*_{i − 1}) × *A*_{i} × *V*(*G*_{i+1}) × ⋯ × *V*(*G*_{n}), *V*(*G*_{1}) × ⋯ × *V*(*G*_{i − 1}) × *B*_{i} × *V*(*G*_{i+1}) × ⋯ × *V*(*G*_{n})} *in which* {*A*_{i}, *B*_{i}} *is a convex cut of* *G*_{i} *for 1* ≤ *i* ≤ *n*.

*Proof*: ⇐= Suppose that *G* = *G*_{1}□⋯□*G*_{n}. If {*A*_{i}, *B*_{i}} is a convex cut of *G*_{i}, then *G*_{i}[*A*_{i}] and *G*_{i}[*B*_{i}] are convex subgraphs of *G*_{i} (1 ≤ *i* ≤ *n*). By Lemma 3.4, *G*[*A*_{i}] = *G*_{1}□⋯□*G*_{i − 1}□*G*_{i}[*A*_{i}]□*G*_{i+1}□⋯□*G*_{n} is a convex subgraph of *G*. Similarly, *G*[*B*_{i}] = *G*_{1}□⋯□*G*_{i − 1}□*G*_{i}[*B*_{i}]□*G*_{i+1}□⋯□*G*_{n} is also a convex subgraph of *G*.

Without loss of generality, suppose that

As {*A*_{i}, *B*_{i}} is a convex cut of *G*_{i} and the vertex *y*_{i} belongs to either *A*_{i} or *B*_{i}, we have that the cut {*V*(*G*[*A*_{i}]), *V*(*G*[*B*_{i}])} = {*A, B*} is a partition of *V*(*G*), and {*A, B*} is a convex cut of *G*.

⇒ Suppose that {*A, B*} is a convex cut of *G*. Then, both *G*[*A*] and *G*[*B*] are convex subgraphs of *G*, and *B* = Ā = *V*(*G*) − *A*. By Lemma 3.4, *G*[*A*] = *G*_{1}[*A*_{1}]□⋯□*G*_{n}[*A*_{n}] and each *G*_{i}[*A*_{i}] is a convex subgraph of *G*_{i} (1 ≤ *i* ≤ *n*).

We now prove that only one *A*_{i} is a proper subset of *V*(*G*_{i}). If there are two proper subsets, without loss of generality, suppose that *A*_{1} is a proper subset of *V*(*G*_{1}), *A*_{2} is that of *V*(*G*_{2}), and *A*_{i} = *G*_{i} (3 ≤ *i* ≤ *n*), *V*(*G*_{j}) − *A*_{j} = *B*_{j} (1 ≤ *j* ≤ *n*). Then, we have that

and

Suppose that *x*_{1} ∈ *A*_{1}, *x*_{2} ∈ *A*_{2}, *y*_{1} ∈ *B*_{1}, *y*_{2} ∈ *B*_{2}, and *x*_{i} ∈ *G*_{i} (3 ≤ *i* ≤ *n*). We have two vertices (*y*_{1}, *x*_{2}, *x*_{3}, *x*_{4}, ..., *x*_{n}) ∈ *B*_{1} × *A*_{2} × *V*(*G*_{3}) × ⋯ × *V*(*G*_{n}) and (*x*_{1}, *y*_{2}, *x*_{3}, *x*_{4}, ..., *x*_{n}) ∈ *A*_{1} × *B*_{2} × *V*(*G*_{3}) × ⋯ × *V*(*G*_{n}). By Lemma 3.5, the distance between them is

However, vertex (*x*_{1}, *x*_{2}, *x*_{3}, *x*_{4}, ..., *x*_{n}) belongs to *A*_{1} × *A*_{2} × *V*(*G*_{3}) × ⋯ × *V*(*G*_{n}), which means that there are two vertices in *B* and a shortest path between them through a vertex in *A*. Therefore, *B* is not a convex subset of *V*(*G*), which contradicts the assertion that {*A, B*} is a convex cut of *G*.

Thus, only one *A*_{i} is a proper subset of *V*(*G*_{i}), and we have that

Similarly, note that *V*(*G*_{j}) − *A*_{j} = *B*_{j} (1 ≤ *j* ≤ *n*), and so

As *G*[*A*] and *G*[*B*] are convex subgraphs of *G*, by Lemma 3.4, both *G*_{i}[*A*_{i}] and *G*_{i}[*B*_{i}] are convex subgraphs of *G*_{i}. Then, *A*_{i} and *B*_{i} are convex subsets of *V*(*G*_{i}), and {*A*_{i}, *B*_{i}} is a convex cut of *G*_{i} (1 ≤ *i* ≤ *n*).

**Proof of Lemma 2.5**. Let *G* be an *l*_{1}-graph and *H* be an isometric subgraph of *G*. By Theorem 2.1, there is a collection ${C}(G)$ such that every edge of *G* is cut by exactly λ cuts.

As *H* is not *l*_{1}-rigid, *H* has another *l*_{1}-embedding. By Theorem 3.2, *G* is an isometric subgraph of the Cartesian product of cocktail party graphs and half-cubes. Let $\widehat{G}={K}_{{m}_{1}\times 2}\square \cdots \square {K}_{{m}_{p}\times 2}\square \frac{1}{2}{Q}_{{n}_{1}}\square \cdots \square \frac{1}{2}{Q}_{{n}_{q}}$ be a Cartesian product that contains *G* as an isometric subgraph, such that each factor of Ĝ is minimal and the number of factors is minimal. Without loss of generality, we assume that *m*_{i} ≤ *m*_{j} and *n*_{i} ≤ *n*_{j} (*i* < *j*).

Because *H* is an isometric subgraph of *G* and *G* is an *l*_{1}-graph, *H* is an *l*_{1}-graph. By Theorem 3.2, *H* has a minimal Cartesian product $\widehat{H}={K}_{{m}_{1}^{\prime}\times 2}\square \cdots \square {K}_{{m}_{s}^{\prime}\times 2}\square \frac{1}{2}{Q}_{{n}_{1}^{\prime}}\square \cdots \square \frac{1}{2}{Q}_{{n}_{t}^{\prime}}$.

As *H* is an isometric subgraph of *G* and *G* is an isometric subgraph of Ĝ, *H* is an isometric subgraph of Ĝ. Because Ĥ may not be equal to Ĝ, we have that *s* ≤ *p*, *t* ≤ *q*, and ${m}_{i}^{\prime}\le {m}_{i}$, ${n}_{j}^{\prime}\le {n}_{j}$ (1 ≤ *i* ≤ *s*, 1 ≤ *j* ≤ *t*).

It is obvious that $\frac{1}{2}{Q}_{{n}_{i}^{\prime}}$ is a convex subgraph of $\frac{1}{2}{Q}_{{n}_{i}}$ (1 ≤ *i* ≤ *t*) and ${K}_{{m}_{i}^{\prime}\times 2}$ is an isometric subgraph of *K*_{mi × 2} (1 ≤ *i* ≤ *s*).

As $\frac{1}{2}{Q}_{n}$ is *l*_{1}-rigid, the collection ${C}(\frac{1}{2}{Q}_{{n}_{i}^{\prime}})$ can expand a collection ${C}(\frac{1}{2}{Q}_{{n}_{i}})$ for 1 ≤ *i* ≤ *t*.

By Theorem 3.3, every collection ${C}({K}_{{m}_{i}^{\prime}\times 2})$ can expand a collection ${C}({K}_{{m}_{i}\times 2})$ (1 ≤ *i* ≤ *s*).

Without loss of generality, suppose that every collection of ${C}({K}_{{m}_{j}\times 2})$ (1 ≤ *j* ≤ *s*) and ${C}(\frac{1}{2}{Q}_{{n}_{k}})$ (1 ≤ *k* ≤ *t*) cuts the edges of the corresponding factors *K*_{mj × 2} and $\frac{1}{2}{Q}_{{n}_{k}}$ exactly λ_{1}, λ_{2}, ..., λ_{s+t} times, respectively. Take the least common multiple λ = [λ_{1}, λ_{2}, ..., λ_{s+t}]. By Lemma 2.2, we have a list of collections ${{C}}^{\prime}({K}_{{m}_{j}\times 2})$ (1 ≤ *j* ≤ *s*) and ${{C}}^{\prime}(\frac{1}{2}{Q}_{{n}_{k}})$ (1 ≤ *k* ≤ *t*) such that every edge of factors *K*_{mj × 2} and $\frac{1}{2}{Q}_{{n}_{k}}$ is cut by exactly λ cuts.

By Theorem 3.6, each convex cut {*A*_{ji}, *B*_{ji}} of ${{C}}^{\prime}({K}_{{m}_{j}\times 2})$ (1 ≤ *j* ≤ *s*) can expand a convex cut {*A, B*} of *G* such that {*p*_{j}(*A*), *p*_{j}(*B*)} = {*A*_{ji}, *B*_{ji}}. This is similar to any convex cut {*A*_{ki}, *B*_{ki}} of ${{C}}^{\prime}(\frac{1}{2}{Q}_{{n}_{k}})$ (1 ≤ *k* ≤ *t*).

All such {*A, B*} expanded by {*A*_{ji}, *B*_{ji}} of ${{C}}^{\prime}({K}_{{m}_{j}\times 2})$ (1 ≤ *j* ≤ *s*) and {*A*_{ki}, *B*_{ki}} of ${{C}}^{\prime}(\frac{1}{2}{Q}_{{n}_{k}})$ (1 ≤ *k* ≤ *t*) form a collection ${C}(G)$ and every edge of *G* is cut by exactly λ cuts of ${C}(G)$. This completes the proof. □

## 4. Conclusion

In this study, we investigated the *l*_{1}-embeddability of the gate-sum graph of two *l*_{1}-graphs. We have shown that the gate-sum graph of two *l*_{1}-graphs *G*_{1} and *G*_{2} is still an *l*_{1}-graph.

## Data Availability Statement

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding author/s.

## Author Contributions

GW contributed the conception of gate-sum of the study. GW and CL contributed to the convex cuts of the gate-sum of two *l*_{1}-graphs. CL and FW organized the literature. FW performed the design of figures. All authors contributed to manuscript revision and read and approved the submitted version.

## Funding

This work was supported by NSFC (Grant Nos. 11861032 and 11961026).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: hypercube, *l*_{1}-embeddability, gate subgraph, gate-sum, convex cuts

Citation: Wang G, Li C and Wang F (2020) *l*_{1}-Embeddability Under Gate-Sum Operation of Two *l*_{1}-Graphs. *Front. Phys.* 8:146. doi: 10.3389/fphy.2020.00146

Received: 22 March 2020; Accepted: 14 April 2020;

Published: 04 June 2020.

Edited by:

Jia-Bao Liu, Anhui Jianzhu University, ChinaReviewed by:

Xiaogang Liu, Northwestern Polytechnical University, ChinaHuiqiu Lin, East China University of Science and Technology, China

Copyright © 2020 Wang, Li and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Guangfu Wang, wgfmath@126.com