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

_{1}-Graphs

^{*}

Edited by: Jia-Bao Liu, Anhui Jianzhu University, China

Reviewed by: Xiaogang Liu, Northwestern Polytechnical University, China; Huiqiu Lin, East China University of Science and Technology, China

This article was submitted to Mathematical Physics, a section of the journal Frontiers in Physics

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.

An _{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 _{1}-embeddability. The subgraph _{1} and _{2} with respect to _{1} and _{2}, such that _{1}∪_{2} = _{1}∩_{2} = _{1} and _{2} are isometric subgraphs of _{1}-graphs is also an _{1}-graph.

_{1}-embeddability

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}_{1}-embeddability. Thus, the purpose of this work is to determine the _{1}-embeddability of the gate-sum graph of two _{1}-graphs.

Let _{G}(_{G}] is a graphic metric space associated with _{H}(_{G}(

Bandelt and Chepoi [

Examples of a convex subgraph

If _{xy} is the path connecting

The _{1}_{1}). Thus, _{1}, _{2}, …) such that _{1}-_{G}) is isometrically embeddable into some _{1}-space. That is, there is a distance-preserving mapping φ from _{G}(_{1}(φ(

The _{n} is the graph whose vertices are ordered

Assouad and Deza [_{1}-graph if and only if _{n} for some positive integers λ and _{n}) such that

λ·_{G}(_{Qn}(ϕ(

for any _{n}, also called a

Shpectorov [_{1}_{n × 2} is a complete multipartite graph with _{2n} deleting a perfect matching, as shown in _{n} is a bipartite graph, and the _{n} is 2.

The complete graph _{4} and the cocktail graph _{4 × 2}.

An _{1}_{1}-graph that essentially admits a unique _{1}-embedding. Shpectorov [_{1}-rigid graph _{1}-_{n} (_{n × 2} (_{1}-rigid, where the variety of _{1}-embeddings of _{n × 2} all come from that of the complete graph _{n}. The half-cube graph _{1}-rigid. Hence, they claim that, if _{1}-rigid, the variety of its _{1}-embeddings arises from that of the complete graph. Deza and Tuma [_{1}-rigid graph. They determined that an _{1}-graph is _{1}-rigid if and only if it is _{4}-free.

Deza and Laurent [_{1}-graphs is also an _{1}-graph. Wang and Zhang [_{1}-graphs along an edge is also an _{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 _{1}-graphs, gluing a convex subgraph cannot guarantee the _{1}-embeddability of the obtained graph. Naturally, we wondered if this result could be generalized.

Suppose that _{i} is a subgraph of _{i}, _{1} is isomorphic to _{2}, their vertices can be identified under some isomorphism as a new graph _{1} and _{2}, denoted by _{1}∪_{H}_{2}. In particular, if _{1}∪_{v}_{2} and _{1}∪_{uv}_{2}, respectively. Additionally, if _{1} and _{2} are isometric in _{1}∪_{H}_{2}, and _{1} and _{2}, then _{1}∪_{H}_{2} is called a _{1} and _{2}, denoted by _{1} and _{2} are isometric subgraphs of _{1}∪_{H}_{2} if and only if _{G1}(_{G2}(

For example, see the graph in _{4} is an isomorphic subgraph of _{1} and _{2}. The _{4}-sum graph _{1}∪_{K4}_{2}, shown in _{4} as the same subgraph. In particular, in _{4} is a gate subgraph of _{2}. Obviously, both _{1} and _{2} are isometric subgraphs of _{1}∪_{K4}_{2}. Therefore, it can be seen as a gate-sum graph _{1} and _{2} with respect to _{4}.

The gate-sum graph _{1} and _{2} with respect to _{4}.

In this paper, we have shown that the gate-sum graph of two _{1}-graphs _{1} and _{2} is also an _{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 _{1}-graphs. We have proven that the collection of convex cuts of the gate-sum graph _{1} and _{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.

Deza and Tuma [_{1}-graphs. A

Deza and Tuma [

For example, in the graph _{4} in _{4} is cut by exactly 2 cuts of {{_{4} is scale-2-embeddable into the hypercube _{4}.

Convex cuts and binary address of _{4}.

Furthermore, Wang and Zhang [_{1}-graph can be proportionally amplified.

Let _{1} and _{2} be two _{1}-graphs and _{1} and _{2}. Without loss of generality, suppose that _{1} is scale-λ-embeddable into some hypercube and _{2} is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections _{1} and _{2} is cut by exactly λ and η cuts, respectively. According to Theorem 2.1 and Lemma 2.2, to prove _{1}-graph, it is sufficient to construct a collection

We now define the

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

_{1}-_{1} _{2}. _{1} _{2})

Next, we will prove that two convex cuts of _{1} and _{2} can expand a convex cut of _{1}, _{1}} of _{1} is cutting _{2}, _{2}} is that of _{2}. Then, {_{1}, _{1}} and {_{2}, _{2}} cut the same edges of _{1}∩_{2} ≠ ∅, then _{1}∩_{2} = ∅. If not, _{1}∩_{2} ≠ ∅ and _{1}∩_{2} ≠ ∅, which contradicts the assertion that {_{1}, _{1}} and {_{2}, _{2}} cut the same edges of _{1}∩_{2} ≠ ∅ and _{1}∩_{2} = ∅. Because _{i}∪_{i} = _{i}) (_{1})∩_{2}) = _{1}∩_{1}∩(_{1}∪_{1})∩(_{2}∪_{2}) = _{1}∩_{2} and _{2}∩_{1}∩_{2}. Similarly, _{1}∩_{1}∩_{2} = _{2}∩_{1})∩_{2}) = (_{1}∪_{1})∩(_{2}∪_{2}) = [_{1}∩(_{2}∪_{2})]∪[_{1}∩(_{2}∪_{2})] = [_{1}∩_{2}]∪[_{1}∩_{2}]. We denote _{A}) = _{1}∩_{2} and _{B}) = _{1}∩_{2}. Then, _{A})∪_{B}) =

_{1}-graphs _{1} and _{2}. Assume that {_{1}, _{1}} is a convex cut of _{1} and {_{2}, _{2}} is that of _{2}. If _{1}-rigid, {_{1}, _{1}} and {_{2}, _{2}} cut the same edges of _{1}, _{1}} and {_{2}, _{2}} can together expand a convex cut_{1}∪_{V(HA)}_{2}, _{1}∪_{V(HB)}_{2}}

If _{1}-rigid, then it has more than one kind of collection of convex cuts. Any two collections

To solve this problem, we have proven that any kind of collection of convex cuts of _{1} and _{2}, respectively, such that they are equal on

_{1}-graph _{1}-rigid, each collection

We will now prove the main theorem of this work.

_{1} _{2}. _{1} _{2} _{1}-_{1}-

_{1}. Because a gate subgraph is a convex subgraph, _{1}. Then, _{1}-graph. Suppose that _{1} is scale-λ-embeddable into some hypercube and _{2} is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections _{1} and _{2} is cut by exactly λ and η cuts, respectively.

If _{1}-rigid,

If _{1}-rigid, the restriction on _{2} is scale-λη-embeddable into some hypercube. Then, _{2} has a collection _{2} is cut by exactly λη cuts. By Lemma 2.5, every _{1} such that every edge of _{1} is cut by exactly λη cuts.

Hence, there always are two collections _{1} and _{2} is cut by exactly λη cuts.

As _{1}, _{1}}, ..., {_{h}, _{h}} and those of _{1}, _{1}}, ..., {_{h}, _{h}} must equal one of _{i}, _{i}} and _{i}, _{i}} and

By Lemma 2.3, the convex cuts of

Now, the convex cuts

Note that, for any graph, a single vertex is a gate subgraph. A _{1} and _{2} are isometric subgraphs of the graphs _{1}∪_{v}_{2} and _{1}∪_{uv}_{2}. The following corollaries can be immediately obtained from Theorem 2.6.

_{1} and _{2} be two _{1}-graphs. _{1}∪_{v}_{2} is an _{1}-graph.

_{1} and _{2} be two _{1}-graphs. If at least one of them is bipartite, _{1}∪_{uv}_{2} is an _{1}-graph.

First, we need the following lemma.

_{1} _{2}. _{1}, _{2}

_{2} is not a convex subgraph of _{1} and _{2} lying in _{2} such that the shortest path _{x1x2} passes through a vertex _{3} of _{1}. As this shortest path must pass through the vertices of the gate subgraph _{1}, there are two vertices _{1}, _{3}-path and _{2}, _{3}-path of _{x1x2}, respectively. Note that both _{1} and _{2} are isometric subgraphs of _{x1x2} is a shortest path

As _{1}, there exists a unique gate _{3} of _{3} in _{2} is a convex subgraph of

_{1}. We need only prove that a convex cut of _{1} or _{2} that does not cut

_{1} that does not cut

Suppose {_{1} that does not cut _{V(H)}_{2})} is a convex cut of

If _{1} and _{2} belonging to _{v1v2} of _{3} of _{H}_{2}. Therefore, _{v1v2} = _{v1v3} + _{v3v2}. If all vertices of _{v1v2} lie entirely in _{1}, _{1}. Without loss of generality, suppose that _{3} lies in _{2}. Note that _{1}. There are two gates _{1} of _{1} and _{2} of _{2} in _{v1v3} and _{v3v2}, respectively. Then, we have that _{v1v2} = _{v1x1} + _{x1v3} + _{v3x2} + _{x2v2}. As _{1} is an isometric subgraph of _{x1x2} that lies entirely in _{1}, and its length equals that of _{x1x2} of _{2}. Then, _{v1v2} = _{v1x1} + _{x1x2} + _{x2v2}, and _{v1v2} lies entirely in _{1}. As _{v1v2} passes through the vertices of _{1}. Therefore,

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

Both _{V(H)}_{2}) are convex sets of _{V(H)}_{2})} is a convex cut of _{V(H)}_{2}), and so the convex cut {_{V(H)}_{2})} is expanded by the convex cut {_{1}.

_{2} that does not cut

Suppose that {_{2} that does not cut _{V(H)}_{1})} is a convex cut of

By Lemma 3.1, it is obvious that _{2} and _{2} a convex subgraph of

Suppose that the vertex set _{V(H)}_{1}) is not a convex set of _{1}, _{2} of _{V(H)}_{1}) such that the shortest path _{v1v2} passes through a vertex _{3} of _{v1v3} and _{v3v2} denote the two segments of _{v1v2} divided by _{3}. Because the vertices _{1}, _{2} belong to _{V(H)}_{1}), we can find two vertices _{1} and _{2} are isometric subgraphs of _{v1v2} is a shortest path, and it passes through the vertex _{3} of _{2}. Therefore, _{V(H)}_{1}) is a convex set of

As _{V(H)}_{1}) are convex sets of _{V(H)}_{1})} is a convex cut of _{2} that does not cut _{V(H)}_{1})} of

□

_{1} and _{2} be two _{1}-graphs and _{1} and _{2}. By Theorem 2.1, there are two collections _{1} and _{2} is cut by exactly λ and η cuts, respectively, as _{1}-rigid, _{1}, _{1}} of _{2}, _{2}} of

Without loss of generality, suppose that _{1}-graph _{1}. Suppose that _{1} of _{1} in _{1}. If _{1} and _{1} belong to different convex sets, assume that _{1} lies in _{1} and _{1} belongs to _{1}∩_{1}∩_{v1u} must pass through the vertices of _{1}, which contradicts the assertion that _{1} is a convex set. Then, both _{1} and _{1} belong to the same convex set _{1} or _{1}.

Without loss of generality, suppose that _{1} and _{1} belong to _{1}. We now show that {_{1} ∪_{V(HA)}_{2}, _{1} ∪ _{V(HB)}_{2}} is a convex cut of _{1} ∪ _{V(HA)}_{2} is a convex set of _{1} and _{2} that belong to _{1} ∪ _{V(HA)}_{2}.

_{1} and _{2} lie in _{2}.

As _{2} is a convex subset of _{2} and _{2} is a convex subgraph of _{2} is a convex subset of _{v1v2} lies entirely in _{2}.

_{1} lies in _{1} and _{2} lies in _{2}.

Because _{1} lies in _{1} and _{2} lies in _{2}, the gate _{1} of _{1} belongs to _{1}∩_{1}, _{1}} and {_{2}, _{2}} cut the same edges of _{1}∩_{2}∩_{1} also belongs to _{2}. Therefore, the shortest path _{v1v2} must pass through the vertices of

If _{v1v2} passes through the gate _{1} of _{1}, we have that _{v1v2} = _{v1x1} + _{x1v2}. Note that both _{1} and _{2} are isometric subgraphs of _{1} and _{1} belong to _{1} and _{1} is a convex set, the path _{v1x1} lies entirely in _{1}. Similarly, _{2} and _{1} belong to _{2}, which is a convex set. Hence, _{v2x1} lies entirely in _{2}. Thus, the shortest path _{v1v2} lies entirely in _{1} ∪ _{V(HA)}_{2}.

If there is a shortest path _{v1v2} that does not pass through the gate _{1} of _{1}, _{v1v2} will pass through a vertex _{3} of _{1}, and _{v1v2} = _{v1x3} + _{x3v2}.

We now prove that _{3} belongs to _{1}∩_{3} lies in _{1}∩_{v1x3} = _{v1x1} + _{x1x3} and _{x1v2} < _{x1x3} + _{x3v2}. Furthermore, _{v1x3} + _{x3v2} = _{v1x1} + _{x1x3} + _{x3v2} > _{v1x1} + _{x1v2}, which contradicts the assertion that _{v1v2} passes through _{3}, but does not pass through the gate _{1}.

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

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

_{1} and _{2} lie in _{1}.

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

Summarizing the above three cases, for any two vertices _{1} and _{2} of _{1} ∪ _{V(HA)}_{2}, we have that the shortest path _{v1v2} lies entirely in _{1} ∪ _{V(HA)}_{2}. It follows that _{1} ∪ _{V(HA)}_{2} is a convex set of

A similar proof shows that the set _{1} ∪ _{V(HB)}_{2} is also a convex set of _{1} ∪ _{V(HA)}_{2}, _{1} ∪ _{V(HB)}_{2}} is a convex cut of _{1}, _{2} and _{1}, _{2}, respectively. Thus, {_{1}, _{1}} of _{1} and {_{2}, _{2}} of _{2} together expand the convex cut {_{1} ∪ _{V(HA)}_{2}, _{1} ∪ _{V(HB)}_{2}} of

To study the expansion of the collection of convex cuts, we have introduced a new characteristic of _{1}-graphs. Shpectorov [_{1}-graphs as follows:

_{1}-graph if and only if it is an isometric subgraph of the Cartesian product of cocktail party graphs and half-cubes

Suppose that _{1}-graph _{1}-graph. By Theorem 3.2, _{1}-rigid, it has a unique collection of convex cuts. Note that

We require the definition of a vertex-transitive graph. An

In other words, a _{1} and _{2} of _{1}) = _{2}.

For a complete graph _{n}, we constructed its collection of convex cuts. Without loss of generality, assume that _{n}) = {_{1}, ..., _{n}}. From Theorem 3.2, _{n} is an _{1}-graph. Suppose that _{n} is scale-λ-embeddable into a hypercube. Theorem 2.1 implies that there is a collection _{n} and λ is even). We assume that {_{1}, _{n}) − _{1}} is a convex cut of _{1} and _{n}) − _{1} are convex sets of _{n}) (|_{1}| = _{i} constructs a convex cut of _{n} of the form _{i} ⊆ _{n}), |_{i}| = _{i}, _{n}) − _{i}}, |_{i}| = _{n} such that every edge of _{n} is cut by the same cuts.

Obviously, there are _{n} has

For _{m × 2} can expand a collection _{n × 2}.

_{n × 2} _{m × 2} _{n × 2}. _{m × 2} _{n × 2}.

_{n × 2} has a complete subgraph _{n}. Without loss of generality, assume that _{n}) = {_{1}, ..., _{n}}, _{n × 2}), then the vertex set

First, we prove that every convex cut of _{n × 2} has only two forms:

Suppose that {_{n × 2}. If _{n × 2}); all vertices of _{n × 2}) will then belong to _{n}) − _{n}) −

Thus, the convex cut of _{n × 2} has only two forms,

Second, we prove that the collection of convex cuts _{i}| = _{n × 2} embeddable into some cubes.

For every edge _{n}, _{n}) − _{i} and _{i}. Note that |_{i}| = _{n}) has

If _{n} and _{i}, or _{n}) − _{i}. Note that

As _{n × 2} is cut by

If _{n × 2}, which only cuts the edges with one end vertex in _{n} and the other one in _{n × 2} is cut by

Let _{i} ⊆ _{n}) such that _{i} ⊆ _{n}) such that _{n × 2}. Obviously, the number of edges with both vertices in _{n}) (or _{n} and the other vertex in _{n × 2} is cut by

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

Similarly, each convex cut of _{m × 2} has only two forms:

Obviously, (_{m}) − _{n}) −

Assume that |_{i}| = |_{j}| is true for all convex cuts of _{i}| = _{n × 2}, in the form _{i}| = _{i}| = _{n × 2} is cut by the same cuts.

Let |_{i}| ≠ |_{j}| for some _{i}| = _{i}| = _{i}| = _{n × 2} such that every edge of _{n × 2} is cut by the same cuts. Similarly, all of the convex cuts _{i}| = _{n × 2} such that every edge of _{n × 2} is cut by the same cuts.

Obviously, the collection _{n × 2} such that every edge of _{n × 2} is cut by the same cuts.

Therefore, every collection _{m × 2} can expand a collection _{n × 2} is cut by the same number of cuts.

We have that, for each cocktail party graph and half-cube, the collection _{1}-graph can expand that of a larger _{1}-graph.

Hammack et al. [

The graphs

_{1}□⋯□_{n} _{1}□⋯□_{n}, _{i} _{i}.

_{1}□⋯□_{n}

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

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

_{1}□⋯□_{n} _{1}) × ⋯ × _{i − 1}) × _{i} × _{i+1}) × ⋯ × _{n}), _{1}) × ⋯ × _{i − 1}) × _{i} × _{i+1}) × ⋯ × _{n})} _{i}, _{i}} _{i}

_{1}□⋯□_{n}. If {_{i}, _{i}} is a convex cut of _{i}, then _{i}[_{i}] and _{i}[_{i}] are convex subgraphs of _{i} (1 ≤ _{i}] = _{1}□⋯□_{i − 1}□_{i}[_{i}]□_{i+1}□⋯□_{n} is a convex subgraph of _{i}] = _{1}□⋯□_{i − 1}□_{i}[_{i}]□_{i+1}□⋯□_{n} is also a convex subgraph of

Without loss of generality, suppose that

As {_{i}, _{i}} is a convex cut of _{i} and the vertex _{i} belongs to either _{i} or _{i}, we have that the cut {_{i}]), _{i}])} = {

⇒ Suppose that {_{1}[_{1}]□⋯□_{n}[_{n}] and each _{i}[_{i}] is a convex subgraph of _{i} (1 ≤

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

and

Suppose that _{1} ∈ _{1}, _{2} ∈ _{2}, _{1} ∈ _{1}, _{2} ∈ _{2}, and _{i} ∈ _{i} (3 ≤ _{1}, _{2}, _{3}, _{4}, ..., _{n}) ∈ _{1} × _{2} × _{3}) × ⋯ × _{n}) and (_{1}, _{2}, _{3}, _{4}, ..., _{n}) ∈ _{1} × _{2} × _{3}) × ⋯ × _{n}). By Lemma 3.5, the distance between them is

However, vertex (_{1}, _{2}, _{3}, _{4}, ..., _{n}) belongs to _{1} × _{2} × _{3}) × ⋯ × _{n}), which means that there are two vertices in

Thus, only one _{i} is a proper subset of _{i}), and we have that

Similarly, note that _{j}) − _{j} = _{j} (1 ≤

As _{i}[_{i}] and _{i}[_{i}] are convex subgraphs of _{i}. Then, _{i} and _{i} are convex subsets of _{i}), and {_{i}, _{i}} is a convex cut of _{i} (1 ≤

_{1}-graph and

As _{1}-rigid, _{1}-embedding. By Theorem 3.2, _{i} ≤ _{j} and _{i} ≤ _{j} (

Because _{1}-graph, _{1}-graph. By Theorem 3.2,

As

It is obvious that _{mi × 2} (1 ≤

As _{1}-rigid, the collection

By Theorem 3.3, every collection

Without loss of generality, suppose that every collection of _{mj × 2} and _{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 _{mj × 2} and

By Theorem 3.6, each convex cut {_{ji}, _{ji}} of _{j}(_{j}(_{ji}, _{ji}}. This is similar to any convex cut {_{ki}, _{ki}} of

All such {_{ji}, _{ji}} of _{ki}, _{ki}} of

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

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

GW contributed the conception of gate-sum of the study. GW and CL contributed to the convex cuts of the gate-sum of two _{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.

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.

_{1}-embedding of weakly median graphs

_{1}-rigid graphs

_{1}-rigid planar graphs

_{1}-embeddable planar graphs

_{1}-embeddability under the edge-gluing operation on graphs