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ORIGINAL RESEARCH article

Front. Phys., 04 June 2020 | https://doi.org/10.3389/fphy.2020.00146

l1-Embeddability Under Gate-Sum Operation of Two l1-Graphs

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

An l1-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 l1-embeddability. The subgraph H of a graph G is called a gate subgraph if, for every vertex vV(G), there exists a vertex xV(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 G1 and G2 with respect to H if H is a gate subgraph of at least one of G1 and G2, such that G1G2 = G, G1G2 = H, and both G1 and G2 are isometric subgraphs of G. In this article, we have shown that the gate-sum graph of two l1-graphs is also an l1-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 l1-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 l1-embeddability. Thus, the purpose of this work is to determine the l1-embeddability of the gate-sum graph of two l1-graphs.

Let G = (V, E) be a connected simple graph. The distance between two vertices u and v of G, denoted by dG(u, v), is the length of a shortest uv path in G. Then [V(G), dG] is a graphic metric space associated with G [3]. A subgraph H of G is an isometric subgraph if dH(u, v) = dG(u, v) for any u, vH. 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 SV(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 vV(G), there exists a unique vertex xV(H) such that x lies on the shortest path between v and any vertex uV(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.

FIGURE 1
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Figure 1. Examples of a convex subgraph (A) and a gate subgraph (B).

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 Pxy 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 l1-space is the metric space of sequences whose series is absolutely convergent, denoted by (X, d1). Thus, X is the set of all real sequences x = (x1, x2, …) such that k=1|xk|<, and the distance function is defined as d1(x,y)=k=1|xk-yk| for any x, yX. A graph G is an l1-graph if (V(G), dG) is isometrically embeddable into some l1-space. That is, there is a distance-preserving mapping φ from V(G) into X such that dG(x, y) = d1(φ(x), φ(y)).

The n-dimensional hypercube Qn 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 l1-graph if and only if G is scale-λ-embeddable into a hypercube Qn for some positive integers λ and n, meaning that there exists a mapping ϕ:V(G) → V(Qn) such that

λ·dG(x, y) = dQn(ϕ(x), ϕ(y))

for any x, yV(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 Qn, also called a partial cube.

Shpectorov [9] and Deza and Grishukhin [10] showed that a graph G is an l1-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 Kn × 2 is a complete multipartite graph with n parts, each of cardinality 2, which is equivalent to a complete graph K2n deleting a perfect matching, as shown in Figure 2. The hypercube Qn is a bipartite graph, and the half-cube 12Qn is the graph defined on one of two parts of this hypercube, with two vertices being joined if the distance between them in Qn is 2.

FIGURE 2
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Figure 2. The complete graph K4 and the cocktail graph K4 × 2.

An l1-rigid graph is an l1-graph that essentially admits a unique l1-embedding. Shpectorov [9] showed that every l1-rigid graph G is an isometric subgraph of a half-cube. He also proved that every l1-rigid graph has scale 1 or 2. Deza and Laurent [11] proved that the complete graph Kn (n ≥ 4) and the cocktail graph Kn × 2 (n ≥ 4) are not l1-rigid, where the variety of l1-embeddings of Kn × 2 all come from that of the complete graph Kn. The half-cube graph 12Qn (n = 3, 4) is l1-rigid. Hence, they claim that, if G is not l1-rigid, the variety of its l1-embeddings arises from that of the complete graph. Deza and Tuma [12] and Chepoi et al. [13] studied the forbidden subgraphs of an l1-rigid graph. They determined that an l1-graph is l1-rigid if and only if it is K4-free.

Deza and Laurent [11] proved that the graph obtained by identifying single vertices from two l1-graphs is also an l1-graph. Wang and Zhang [14] proved that the graph obtained by gluing two l1-graphs along an edge is also an l1-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 l1-graphs, gluing a convex subgraph cannot guarantee the l1-embeddability of the obtained graph. Naturally, we wondered if this result could be generalized.

Suppose that Hi is a subgraph of Gi, i = 1, 2. If H1 is isomorphic to H2, 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 G1 and G2, denoted by G1HG2. 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 G1vG2 and G1uvG2, respectively. Additionally, if G1 and G2 are isometric in G1HG2, and H is a gate subgraph of at least one of G1 and G2, then G1HG2 is called a gate-sum of G1 and G2, denoted by G1HgG2. Both G1 and G2 are isometric subgraphs of G1HG2 if and only if dG1(x, y) = dG2(x, y) for any x, yH.

For example, see the graph in Figure 3, where the marked K4 is an isomorphic subgraph of G1 and G2. The K4-sum graph G1K4G2, shown in Figure 3C, is obtained by identifying these two marked K4 as the same subgraph. In particular, in Figure 3B, the marked K4 is a gate subgraph of G2. Obviously, both G1 and G2 are isometric subgraphs of G1K4G2. Therefore, it can be seen as a gate-sum graph G1K4gG2 of G1 and G2 with respect to K4.

FIGURE 3
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Figure 3. The gate-sum graph G1K4gG2 of G1 and G2 with respect to K4.

In this paper, we have shown that the gate-sum graph of two l1-graphs G1 and G2 is also an l1-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 l1-graphs. We have proven that the collection of convex cuts of the gate-sum graph G1HgG2 can be expanded by those of G1 and G2. 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 l1-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 uA and vB. An edge cut of G is a subset of E(G) of the form [S,S¯], where S is a nonempty proper subset of V(G), S¯=V\S, and [S,S¯] is the set of edges with one end in S and the other in S¯. Similarly, we say that a cut {A, B} of G cuts a subgraph H if [AV(H), BV(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 K4 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 K4 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 K4 is scale-2-embeddable into the hypercube Q4.

FIGURE 4
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Figure 4. Convex cuts and binary address of K4.

Furthermore, Wang and Zhang [14] showed that the scale of an l1-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-rλ-embeddable into a hypercube.

Let G1 and G2 be two l1-graphs and G1HgG2 be a gate-sum graph of G1 and G2. Without loss of generality, suppose that G1 is scale-λ-embeddable into some hypercube and G2 is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections C(G1) and C(G2) such that every edge of G1 and G2 is cut by exactly λ and η cuts, respectively. According to Theorem 2.1 and Lemma 2.2, to prove G1HgG2 is an l1-graph, it is sufficient to construct a collection C(G1HgG2) of convex cuts of G1HgG2 such that every edge of G1HgG2 is cut by exactly the same number of cuts. Now, we construct a collection of convex cuts of G1HgG2 from the convex cuts of C(G1) and C(G2).

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 AA′ and BB′, 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 G1HgG2 is a gate-sum graph of two l1-graphs G1 and G2. Then, a convex cut of G1 (or G2) not cutting H can expand a convex cut of G1HgG2.

Next, we will prove that two convex cuts of G1 and G2 can expand a convex cut of G1HgG2 if they cut the same edges of H. Suppose that the convex cut {A1, B1} of G1 is cutting H and that the cut {A2, B2} is that of G2. Then, {A1, B1} and {A2, B2} cut the same edges of H. If A1A2 ≠ ∅, then A1B2 = ∅. If not, A1A2 ≠ ∅ and A1B2 ≠ ∅, which contradicts the assertion that {A1, B1} and {A2, B2} cut the same edges of H. Similarly, we have B1B2 ≠ ∅ and B1A2 = ∅. Because AiBi = V(Gi) (i = 1, 2) and V(G1)∩V(G2) = V(H), we know that A1V(H) = A1∩(A1B1)∩(A2B2) = A1A2 and A2V(H) = A1A2. Similarly, B1V(H) = B1B2 = B2V(H). Furthermore, we have that V(H) = V(G1)∩V(G2) = (A1B1)∩(A2B2) = [A1∩(A2B2)]∪[B1∩(A2B2)] = [A1A2]∪[B1B2]. We denote V(HA) = A1A2 and V(HB) = B1B2. Then, V(HA)∪V(HB) = V(H), and we have the following lemma.

Lemma 2.4. Suppose that G1HgG2 is a gate-sum graph of two l1-graphs G1 and G2. Assume that {A1, B1} is a convex cut of G1 and {A2, B2} is that of G2. If H is l1-rigid, {A1, B1} and {A2, B2} cut the same edges of H. Then, {A1, B1} and {A2, B2} can together expand a convex cut {A1V(HA)A2, B1V(HB)B2} of G1HgG2.

If H is not l1-rigid, then it has more than one kind of collection of convex cuts. Any two collections C(G1) and C(G2) may not be equal on H. Therefore, the convex cuts of C(G1) and C(G2) 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 G1 and G2, respectively, such that they are equal on H.

Lemma 2.5. Let H be an isometric subgraph of an l1-graph G. If H is not l1-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 G1HgG2 is a gate-sum graph of G1 and G2. If G1 and G2 are l1-embeddable, then G1HgG2 is also l1-embeddable.

Proof: Without loss of generality, suppose that H is a gate subgraph of G1. Because a gate subgraph is a convex subgraph, H is a convex subgraph of G1. Then, H is an l1-graph. Suppose that G1 is scale-λ-embeddable into some hypercube and G2 is scale-η-embeddable into some hypercube. By Theorem 2.1, there are two collections C(G1) and C(G2) such that every edge of G1 and G2 is cut by exactly λ and η cuts, respectively.

If H is l1-rigid, H has only one kind of collection of convex cuts. Then, C(G1) and C(G2) have the same restriction on H (which means that λ = η).

If H is not l1-rigid, the restriction on H of C(G1) is not equal to that of C(G2). Suppose that λ ≠ η. By Lemma 2.2, G2 is scale-λη-embeddable into some hypercube. Then, G2 has a collection C(G2) such that every edge of G2 is cut by exactly λη cuts. By Lemma 2.5, every C(H) can expand a collection C(G1). Obviously, the restriction on H of C(G2) is a kind of C(H). Thus, it can expand a new collection C(G1) of G1 such that every edge of G1 is cut by exactly λη cuts.

Hence, there always are two collections C(G1) and C(G2) for which the restrictions of them on H are equal, and every edge of G1 and G2 is cut by exactly λη cuts.

As C(G1) and C(G2) are equal on H, there are the same number of convex cuts of C(G1) and C(G2) cutting H. Denote the convex cuts of C(G1) that are cutting H as {A1, B1}, ..., {Ah, Bh} and those of C(G2) as {A1,B1},...,{Ah,Bh}. Because the restrictions on H of C(G1) and C(G2) are equal, each convex cut of {A1, B1}, ..., {Ah, Bh} must equal one of {A1,B1},...,{Ah,Bh} on H. Without loss of generality, we assume that each pair of {Ai, Bi} and {Ai,Bi} cut the same edges of H (1 ≤ ih). By Lemma 2.4, each pair of convex cuts {Ai, Bi} and {Ai,Bi} can together expand a convex cut {AiAi,BiBi} of G1HgG2 (1 ≤ ih). Then, every edge of H is cut by {AiAi,BiBi} to give exactly λη cuts (1 ≤ ih).

By Lemma 2.3, the convex cuts of C(G1) and C(G2) that do not cut H can expand the convex cuts of G1HgG2 that do not cut H.

Now, the convex cuts {AiAi,BiBi} for 1 ≤ ih, together with the convex cuts of C(G1) and C(G2) that do not cut H, form a collection of convex cuts of G1HgG2, such that every edge of G1HgG2 is cut by λη convex cuts. Therefore, by Theorem 2.1, the graph G1HgG2 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 G1 and G2 are isometric subgraphs of the graphs G1vG2 and G1uvG2. The following corollaries can be immediately obtained from Theorem 2.6.

Corollary 2.7. ([11]). Let G1 and G2 be two l1-graphs. G1vG2 is an l1-graph.

Corollary 2.8. ([14]). Let G1 and G2 be two l1-graphs. If at least one of them is bipartite, G1uvG2 is an l1-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 G1HgG2 is a gate-sum graph of G1 and G2. If H is a gate subgraph of G1, then G2 is a convex subgraph of G1HgG2.

Proof: If G2 is not a convex subgraph of G1HgG2, there are two vertices x1 and x2 lying in G2 such that the shortest path Px1x2 passes through a vertex v3 of G1. As this shortest path must pass through the vertices of the gate subgraph H of G1, there are two vertices x1 and x2 of H on the x1, v3-path and x2, v3-path of Px1x2, respectively. Note that both G1 and G2 are isometric subgraphs of G1HgG2. It is clear that the segment from x1 to x2 of Px1x2 is a shortest path Px1x2. Then, we have Px1x2=Px1v3+Pv3x2.

As H is a gate subgraph of G1, there exists a unique gate a3 of v3 in H such that Px1v3=Px1a3+Pa3v3 and Px2v3=Px2a3+Pa3v3. Then, we have that Px1x2=Px1v3+Pv3x2=Px1a3+Pa3v3+Px2a3+Pa3v3>Px1a3+Pa3x2, which contradicts the assertion that Px1x2 is a shortest path. Thus, G2 is a convex subgraph of G1HgG2.

Proof of Lemma 2.3. Without loss of generality, suppose that H is a gate subgraph of G1. We need only prove that a convex cut of G1 or G2 that does not cut H can expand a convex cut of G1HgG2.

Case 1. A convex cut of G1 that does not cut H can expand that of G1HgG2.

Suppose {A, B} is a convex cut of G1 that does not cut H. Without loss of generality, we assume that V(H) ⊆ B. We now prove that {A, BV(H)V(G2)} is a convex cut of G1HgG2 that does not cut H and is expanded by {A, B}.

If A is not a convex set of G1HgG2, there are two vertices v1 and v2 belonging to A such that Pv1v2 of G1HgG2 passes through a vertex v3 of G[B]∪HG2. Therefore, Pv1v2 = Pv1v3 + Pv3v2. If all vertices of Pv1v2 lie entirely in G1, A cannot be a convex set of G1. Without loss of generality, suppose that v3 lies in G2. Note that H is a gate subgraph of G1. There are two gates x1 of v1 and x2 of v2 in H, and these two gates lie in Pv1v3 and Pv3v2, respectively. Then, we have that Pv1v2 = Pv1x1 + Px1v3 + Pv3x2 + Px2v2. As G1 is an isometric subgraph of G1HgG2, there is some Px1x2 that lies entirely in G1, and its length equals that of Px1x2 of G2. Then, Pv1v2 = Pv1x1 + Px1x2 + Px2v2, and Pv1v2 lies entirely in G1. As Pv1v2 passes through the vertices of H, and H belongs to B, A cannot be a convex set of G1. Therefore, A is a convex set of G1HgG2.

If BV(H)V(G2) is not a convex set of G1HgG2, there are two vertices v4 and v5 belonging to BV(H)V(G2) such that Pv4v5 passes through a vertex v6 in A and Pv4v5 = Pv4v6 + Pv6v5. Obviously, the path Pv4v6 does not intersect with Pv6v5 at any internal vertices. The segment of a shortest path is still a shortest path. This means that v6 has two internally disjoint paths Pv6v4 and Pv6v5 that connect with the vertices in H. Thus, v6 has at least two gates, which contradicts the statement that the gate is unique.

Both A and BV(H)V(G2) are convex sets of G1HgG2, and they contain all vertices of G1HgG2. Thus, {A, BV(H)V(G2)} is a convex cut of G1HgG2. Furthermore, note that A = A and BBV(H)V(G2), and so the convex cut {A, BV(H)V(G2)} is expanded by the convex cut {A, B} of G1.

Case 2. A convex cut of G2 that does not cut H can expand that of G1HgG2.

Suppose that {C, D} is a convex cut of G2 that does not cut H. Without loss of generality, we assume that HD. We now prove that the cut {C, DV(H)V(G1)} is a convex cut of G1HgG2 and is expanded by {C, D}.

By Lemma 3.1, it is obvious that C is a convex set of G1HgG2 because C is a convex set of G2 and G2 a convex subgraph of G1HgG2.

Suppose that the vertex set DV(H)V(G1) is not a convex set of G1HgG2. There will be two vertices v1, v2 of DV(H)V(G1) such that the shortest path Pv1v2 passes through a vertex v3 of C. Let Pv1v3 and Pv3v2 denote the two segments of Pv1v2 divided by v3. Because the vertices v1, v2 belong to DV(H)V(G1), we can find two vertices v1,v2 of D such that v1Pv1v3 and v2Pv3v2. Note that both G1 and G2 are isometric subgraphs of G1HgG2. It is clear that the segment from v1 to v2 of the path Pv1v2 is a shortest path, and it passes through the vertex v3 of C, which contradicts the assertion that D is a convex set of G2. Therefore, DV(H)V(G1) is a convex set of G.

As C and DV(H)V(G1) are convex sets of G1HgG2, {C, DV(H)V(G1)} 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 G2 that does not cut H expands the convex cut {C, DV(H)V(G1)} of G1HgG2. This completes the proof.

3.2. Proof of Lemma 2.4

Proof: Let G1 and G2 be two l1-graphs and G1HgG2 be the gate-sum graph of G1 and G2. By Theorem 2.1, there are two collections C(G1) and C(G2) such that every edge of G1 and G2 is cut by exactly λ and η cuts, respectively, as H is l1-rigid, C(G1) and C(G2) must be equal on H. For any convex cut {A1, B1} of C(G1), we can find a convex cut {A2, B2} of C(G2) that cuts the same edge of H.

Without loss of generality, suppose that H is a gate subgraph of an l1-graph G1. Suppose that x1 of V(H) is the gate of v1 in G1. If v1 and x1 belong to different convex sets, assume that v1 lies in A1 and x1 belongs to B1V(H). There will be a vertex u in A1V(H) such that the shortest path Pv1u must pass through the vertices of B1, which contradicts the assertion that A1 is a convex set. Then, both v1 and x1 belong to the same convex set A1 or B1.

Without loss of generality, suppose that v1 and x1 belong to A1. We now show that {A1V(HA)A2, B1V(HB)B2} is a convex cut of G1HgG2. First, we prove that A1V(HA)A2 is a convex set of G1HgG2. Consider two vertices v1 and v2 that belong to A1V(HA)A2.

Case 1. Both v1 and v2 lie in A2.

As A2 is a convex subset of G2 and G2 is a convex subgraph of G1HgG2, A2 is a convex subset of G1HgG2. Obviously, Pv1v2 lies entirely in A2.

Case 2. The vertex v1 lies in A1 and v2 lies in A2.

Because v1 lies in A1 and v2 lies in A2, the gate x1 of v1 belongs to A1V(H). As {A1, B1} and {A2, B2} cut the same edges of H, we have that A1V(H) = A2V(H) and x1 also belongs to A2. Therefore, the shortest path Pv1v2 must pass through the vertices of H.

If Pv1v2 passes through the gate x1 of v1, we have that Pv1v2 = Pv1x1 + Px1v2. Note that both G1 and G2 are isometric subgraphs of G1HgG2. As both v1 and x1 belong to A1 and A1 is a convex set, the path Pv1x1 lies entirely in A1. Similarly, v2 and x1 belong to A2, which is a convex set. Hence, Pv2x1 lies entirely in A2. Thus, the shortest path Pv1v2 lies entirely in A1V(HA)A2.

If there is a shortest path Pv1v2 that does not pass through the gate x1 of v1, Pv1v2 will pass through a vertex x3 of V(H), which is not the gate of v1, and Pv1v2 = Pv1x3 + Px3v2.

We now prove that x3 belongs to A1V(H). If this is not the case, then x3 lies in B1V(H), and so Pv1x3 = Pv1x1 + Px1x3 and Px1v2 < Px1x3 + Px3v2. Furthermore, Pv1x3 + Px3v2 = Pv1x1 + Px1x3 + Px3v2 > Pv1x1 + Px1v2, which contradicts the assertion that Pv1v2 passes through x3, but does not pass through the gate x1.

As v1 and x3 belong to A1, and x3 and v2 belong to A2, we have that Pv1x3 lies entirely in A1 and Px3v2 lies entirely in A2. Therefore, Pv1v2 = Pv1x3 + Px3v2 lies entirely in A1V(HA)A2.

Hence, for any vertex v1 of A1 and any vertex v2 of A2, Pv1v2 lies entirely in A1V(HA)A2. This proves case 2.

Case 3. Both v1 and v2 lie in A1.

If Pv1v2 does not pass through the vertices of G2, then Pv1v2 lies in G1. Note that A1 is a convex subgraph of G1, and Pv1v2 lies in A1. If Pv1v2 passes through the vertices of G2, it must pass through a vertex v3 of A2. From case 2, we know that both Pv1v3 and Pv3v2 lie in A1V(HA)A2 and that Pv1v2 lies entirely in A1V(HA)A2.

Summarizing the above three cases, for any two vertices v1 and v2 of A1V(HA)A2, we have that the shortest path Pv1v2 lies entirely in A1V(HA)A2. It follows that A1V(HA)A2 is a convex set of G1HgG2.

A similar proof shows that the set B1V(HB)B2 is also a convex set of G1HgG2. Then, {A1V(HA)A2, B1V(HB)B2} is a convex cut of G1HgG2, and its two convex sets contain vertex sets A1, A2 and B1, B2, respectively. Thus, {A1, B1} of G1 and {A2, B2} of G2 together expand the convex cut {A1V(HA)A2, B1V(HB)B2} of G1HgG2.

3.3. Proof of Lemma 2.5

To study the expansion of the collection of convex cuts, we have introduced a new characteristic of l1-graphs. Shpectorov [9] and Deza and Grishukhin [10] characterized l1-graphs as follows:

Theorem 3.2. ([9, 10]) A graph G is an l1-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 l1-graph G; H is also an l1-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 l1-rigid, it has a unique collection of convex cuts. Note that 12Qm is a subgraph of 12Qn. Thus, we have that any collection C(12Qm) of 12Qm can expand a collection C(12Qn) of 12Qn (mn). We need only examine whether any collection C(Km×2) can expand a collection C(Kn×2) (mn).

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 v1 and v2 of G, there is some automorphism f:V(G) → V(G) such that f(v1) = v2.

For a complete graph Kn, we constructed its collection of convex cuts. Without loss of generality, assume that V(Kn) = {v1, ..., vn}. From Theorem 3.2, Kn is an l1-graph. Suppose that Kn is scale-λ-embeddable into a hypercube. Theorem 2.1 implies that there is a collection C(Kn) such that every edge uv is cut by λ cuts (u, v belong to Kn and λ is even). We assume that {S1, V(Kn) − S1} is a convex cut of C(Kn), and that both S1 and V(Kn) − S1 are convex sets of V(Kn) (|S1| = q). As the complete graph is vertex-transitive, each Si constructs a convex cut of Kn of the form SiV(Kn), |Si| = q ((1i(nq))). Then, we have that all convex cuts {Si, V(Kn) − Si}, |Si| = q (1i(nq)), form a collection of convex cuts of Kn such that every edge of Kn is cut by the same cuts.

Obviously, there are (nq) different convex cuts, and each convex cut acts on q(nq) edges. Note that the complete graph Kn has n(n-1)2 edges and is vertex-transitive. Thus, we have that λ=(nq)q(n-q)n(n-1)2=2(n-2q-1).

For mn, we can now prove that the collection C(Km×2) of Km × 2 can expand a collection C(Kn×2) of Kn × 2.

Theorem 3.3. Let Kn × 2 be a cocktail party graph and Km × 2 be a cocktail party subgraph of Kn × 2. Every collection C(Km×2) of Km × 2 can expand a collection C(Kn×2) of Kn × 2.

Proof: Obviously, the cocktail party graph Kn × 2 has a complete subgraph Kn. Without loss of generality, assume that V(Kn) = {v1, ..., vn}, V(Kn)={v1,...,vn}, and V(Kn×2)={v1,...,vn,v1,...,vn} such that dKn×2(vj,vj)=2 (1 ≤ jn), dKn×2(vi,vj)=dKn×2(vi,vj)=1 (ij). If the vertex set S is a subset of V(Kn × 2), then the vertex set S′ = {x′|xS} is a subset of V(Kn×2).

First, we prove that every convex cut of Kn × 2 has only two forms: {S(V(Kn)-S),S(V(Kn)-S)} and {V(Kn),V(Kn)}.

Suppose that {A, B} is a convex cut of Kn × 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(Kn × 2); all vertices of V(Kn × 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 SA and S′ ⊆ B. If V(Kn) − SA, then V(Kn)-SB and {A,B}={V(Kn),V(Kn)}. If V(Kn) − SB, then V(Kn)-SA and {A,B}={S(V(Kn)-S), S(V(Kn)-S)}.

Thus, the convex cut of Kn × 2 has only two forms, {S(V(Kn)-S),S(V(Kn)-S)} and {V(Kn),V(Kn)}.

Second, we prove that the collection of convex cuts {Si(V(Kn)-Si),Si(V(Kn)-Si)}, |Si| = q ((1i(nq))), together with some {V(Kn),V(Kn)} make the cocktail graph Kn × 2 embeddable into some cubes.

For every edge uv in Kn, uv is cut by the convex cut {Si(V(Kn)-Si),Si(V(Kn)-Si)}. We have that u(Si(V(Kn)-Si))V(Kn)=Si and v(Si(V(Kn)-Si))V(Kn)=V(Kn)-Si, or uV(Kn) − Si and vSi. Note that |Si| = q and V(Kn) has n vertices, so the number of convex cuts that cut edge uv is 2(n-2q-1). This is similar to each edge uv′ of Kn.

If uKn and vKn, uv′ is cut by the convex cut {Si(V(Kn)-Si),Si(V(Kn)-Si)}. We have that u(Si(V(Kn)-Si))V(Kn)=Si, v(Si{V(Kn)-Si})V(Kn)=Si, uSi, and vSi, or u(Si{V(Kn)-Si})V(Kn)=V(Kn)-Si, v(Si(V(Kn)-Si))V(Kn)=V(Kn)-Si, uV(Kn)-Si, and vV(Kn) − Si. Note that |Si|=|Si|=q and |V(Kn)-Si|=|V(Kn)-Si|=n-q, so the number of convex cuts that cut edge uv′ is (n-2q-2)+(n-2n-q-2)=(n-2q-2)+(n-2q).

As n-n2qn+n2, 2(n-2q-1)(n-2q-2)+(n-2q). If 2(n-2q-1)=(n-2q-2)+(n-2q), every edge of Kn × 2 is cut by 2(n-2q-1) cuts.

If 2(n-2q-1)>(n-2q-2)+(n-2q), then n-n2<q<n+n2. Obviously, {V(Kn),V(Kn)} is a convex cut of Kn × 2, which only cuts the edges with one end vertex in Kn and the other one in Kn. Then, the collection {Si(V(Kn)-Si),Si(V(Kn)-Si)} ((1i(nq))) together with {V(Kn),V(Kn)} form a new collection C(Kn×2) such that every edge of Kn × 2 is cut by 2(n-2q-1) cuts.

Let 2(n-2q-1)<(n-2q-2)+(n-2q). If n is even, choose TiV(Kn) such that |Ti|=n2 ((1i(nn2)). If n is odd, choose TiV(Kn) such that |Ti|=n+12 (1i(nn+12)). Then, {Ti(V(Kn)-Ti),Ti(V(Kn)-Ti)} is a convex cut of Kn × 2. Obviously, the number of edges with both vertices in V(Kn) (or V(Kn)) that are cut by {Ti(V(Kn)-Ti),Ti(V(Kn)-Ti)} is greater than the number of edges that are cut by the same cut with one end vertex in Kn and the other vertex in Kn. Thus, the collection {Si(V(Kn)-Si),Si(V(Kn)-Si)} ((1i(nq))) together with {Ti(V(Kn)-Ti), Ti(V(Kn)-Ti)} (1i(nn2) (or(nn+12)))) and {V(Kn),V(Kn)} form a new collection C(Kn×2) such that every edge of Kn × 2 is cut by 2(n-2q-1)+2a(n-2n2-1) cuts. The constant a is the minimal number such that 2(n-2q-1)+2a(n-2n2-1)(n-2q-2)+(n-2q)+a((n-2n2-2)+(n-2n2)).

Third, we prove that every collection of convex cuts of Km × 2 can expand that of Kn × 2 (mn).

Similarly, each convex cut of Km × 2 has only two forms: {A(V(Km)-A),A(V(Km)-A)}, and {V(Km),V(Km)}.

Obviously, (V(Km) − A) ⊆ (V(Kn) − A) and (V(Km)-A)(V(Kn)-A). Then, each convex cut {A(V(Km)-A),A(V(Km)-A)} of C(Km×2) can expand a convex cut {A(V(Kn)-A),A(V(Kn)-A)} of C(Kn×2). Similarly, the convex cut {V(Km),V(Km)} expands the cut {V(Kn),V(Kn)}.

Assume that |Ai| = |Aj| is true for all convex cuts of C(Km×2) except the convex cut {V(Km),V(Km)}. This means that {Ai(V(Km)-Ai),Ai(V(Km)-Ai)}, |Ai| = q ((1i(mq))). Then, all of the cuts together with {V(Km),V(Km)} expand a collection of convex cuts of Kn × 2, in the form {Ai(V(Kn)-Ai),Ai(V(Kn)-Ai)}, |Ai| = q ((1i(nq))), together with {V(Kn),V(Kn)}. By the second part, {Ai(V(Kn)-Ai),Ai(V(Kn)-Ai)}, |Ai| = q ((1i(nq))), together with {V(Kn),V(Kn)} ensure that every edge of the graph Kn × 2 is cut by the same cuts.

Let |Ai| ≠ |Aj| for some i and j of the convex cuts of C(Km×2). Without loss of generality, suppose that C(Km×2) has three kinds of convex cuts, formed as {Ai(V(Kn)-Ai),Ai(V(Kn)-Ai)}, |Ai| = q ((1i(nq))), and {Bi(V(Kn)-Bi),Bi(V(Kn)-Bi)}, |Bi| = p ((1i(np))), together with {V(Kn),V(Kn)}. By the above discussion, all of the convex cuts {Ai(V(Kn)-Ai),Ai(V(Kn)-Ai)}, |Ai| = q ((1i(nq))), together with {V(Kn),V(Kn)} expand a collection C1(Kn×2) of convex cuts of Kn × 2 such that every edge of Kn × 2 is cut by the same cuts. Similarly, all of the convex cuts {Bi(V(Kn)-Bi),Bi(V(Kn)-Bi)}, |Bi| = p ((1i(np)), together with {V(Kn),V(Kn)} expand a collection C2(Kn×2) of convex cuts of Kn × 2 such that every edge of Kn × 2 is cut by the same cuts.

Obviously, the collection C1(Kn×2) together with the collection C2(Kn×2) is still a collection of convex cuts of Kn × 2 such that every edge of Kn × 2 is cut by the same cuts.

Therefore, every collection C(Km×2) of Km × 2 can expand a collection C(Kn×2) such that every edge of Kn × 2 is cut by the same number of cuts.

We have that, for each cocktail party graph and half-cube, the collection C(12Qm) can expand a collection C(12Qn), and the collection C(Km×2) can expand a collection C(Kn×2) (mn). By Theorem 3.2, we can prove that the collection of convex cuts of an l1-graph can expand that of a larger l1-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)|uV(G) and vV(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 = G1□⋯□Gn is convex if and only if W = W1□⋯□Wn, where each Wi is convex in Gi.

Lemma 3.5. ([6]) If G = G1□⋯□Gn and x, yV(G), then

dG(x,y)=i=1ndGi(pi(x),pi(y))

For any index 1 ≤ in, pi is a projection map pi:G1□⋯□GnGi, defined as pi(x1, x2, ..., xn) = xi.

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 = G1□⋯□Gn if and only if {A, B} has the form {V(G1) × ⋯ × V(Gi − 1) × Ai × V(Gi+1) × ⋯ × V(Gn), V(G1) × ⋯ × V(Gi − 1) × Bi × V(Gi+1) × ⋯ × V(Gn)} in which {Ai, Bi} is a convex cut of Gi for 1in.

Proof: ⇐= Suppose that G = G1□⋯□Gn. If {Ai, Bi} is a convex cut of Gi, then Gi[Ai] and Gi[Bi] are convex subgraphs of Gi (1 ≤ in). By Lemma 3.4, G[Ai] = G1□⋯□Gi − 1Gi[Ai]□Gi+1□⋯□Gn is a convex subgraph of G. Similarly, G[Bi] = G1□⋯□Gi − 1Gi[Bi]□Gi+1□⋯□Gn is also a convex subgraph of G.

Without loss of generality, suppose that

V(G)={(x1,...,xi,...,xn)|xiV(Gi)}V(G[Ai])={(x1,...,xi-1,yi,xi+1,...,xn)|xjV(Gj),ji,yiAi}={V(G1)××V(Gi-1)×Ai×V(Gi+1)××V(Gn)}V(G[Bi])={(x1,...,xi-1,yi,xi+1,...,xn)|xjV(Gj),ji,yiBi}={V(G1)××V(Gi-1)×Bi×V(Gi+1)××V(Gn)}.

As {Ai, Bi} is a convex cut of Gi and the vertex yi belongs to either Ai or Bi, we have that the cut {V(G[Ai]), V(G[Bi])} = {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] = G1[A1]□⋯□Gn[An] and each Gi[Ai] is a convex subgraph of Gi (1 ≤ in).

We now prove that only one Ai is a proper subset of V(Gi). If there are two proper subsets, without loss of generality, suppose that A1 is a proper subset of V(G1), A2 is that of V(G2), and Ai = Gi (3 ≤ in), V(Gj) − Aj = Bj (1 ≤ jn). Then, we have that

A={(x1,x2,...,xn)|xiV(Gi),i1,2,x1A1,x2A2}   ={A1×A2×V(G3)××V(Gn)}

and

Ā=B={(x1,x2,...,xn)|xiV(Gi),i1,2,x1A1,x2A2,                or x1A1,x2A2, or x1A1,x2A2}           ={[(B1×A2)(A1×B2)(B1×B2)]×V(G3)               ××V(Gn)}.

Suppose that x1A1, x2A2, y1B1, y2B2, and xiGi (3 ≤ in). We have two vertices (y1, x2, x3, x4, ..., xn) ∈ B1 × A2 × V(G3) × ⋯ × V(Gn) and (x1, y2, x3, x4, ..., xn) ∈ A1 × B2 × V(G3) × ⋯ × V(Gn). By Lemma 3.5, the distance between them is

dG((y1,x2,x3,x4,...,xn),(x1,y2,x3,x4,...,xn))=dG1(y1,x1)    +dG2(x2,y2)    =dG((y1,x2,x3,x4,...,xn),(x1,x2,x3,x4,...,xn))    +dG((x1,x2,x3,x4,...,xn),(x1,y2,x3,x4,...,xn)).

However, vertex (x1, x2, x3, x4, ..., xn) belongs to A1 × A2 × V(G3) × ⋯ × V(Gn), 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 Ai is a proper subset of V(Gi), and we have that

A={(x1,...,xi-1,yi,xi+1,...,xn)|xjV(Gj),ji,yiAi}    ={V(G1)××V(Gi-1)×Ai×V(Gi+1)××V(Gn)}.

Similarly, note that V(Gj) − Aj = Bj (1 ≤ jn), and so

B={(x1,...,xi-1,yi,xi+1,...,xn)|xjV(Gj),ji,yiAi}    ={V(G1)××V(Gi-1)×Bi×V(Gi+1)××V(Gn)}.

As G[A] and G[B] are convex subgraphs of G, by Lemma 3.4, both Gi[Ai] and Gi[Bi] are convex subgraphs of Gi. Then, Ai and Bi are convex subsets of V(Gi), and {Ai, Bi} is a convex cut of Gi (1 ≤ in).

Proof of Lemma 2.5. Let G be an l1-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 l1-rigid, H has another l1-embedding. By Theorem 3.2, G is an isometric subgraph of the Cartesian product of cocktail party graphs and half-cubes. Let G^=Km1×2Kmp×212Qn112Qnq 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 mimj and ninj (i < j).

Because H is an isometric subgraph of G and G is an l1-graph, H is an l1-graph. By Theorem 3.2, H has a minimal Cartesian product H^=Km1×2Kms×212Qn112Qnt.

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 sp, tq, and mimi, njnj (1 ≤ is, 1 ≤ jt).

It is obvious that 12Qni is a convex subgraph of 12Qni (1 ≤ it) and Kmi×2 is an isometric subgraph of Kmi × 2 (1 ≤ is).

As 12Qn is l1-rigid, the collection C(12Qni) can expand a collection C(12Qni) for 1 ≤ it.

By Theorem 3.3, every collection C(Kmi×2) can expand a collection C(Kmi×2) (1 ≤ is).

Without loss of generality, suppose that every collection of C(Kmj×2) (1 ≤ js) and C(12Qnk) (1 ≤ kt) cuts the edges of the corresponding factors Kmj × 2 and 12Qnk exactly λ1, λ2, ..., λs+t times, respectively. Take the least common multiple λ = [λ1, λ2, ..., λs+t]. By Lemma 2.2, we have a list of collections C(Kmj×2) (1 ≤ js) and C(12Qnk) (1 ≤ kt) such that every edge of factors Kmj × 2 and 12Qnk is cut by exactly λ cuts.

By Theorem 3.6, each convex cut {Aji, Bji} of C(Kmj×2) (1 ≤ js) can expand a convex cut {A, B} of G such that {pj(A), pj(B)} = {Aji, Bji}. This is similar to any convex cut {Aki, Bki} of C(12Qnk) (1 ≤ kt).

All such {A, B} expanded by {Aji, Bji} of C(Kmj×2) (1 ≤ js) and {Aki, Bki} of C(12Qnk) (1 ≤ kt) 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 l1-embeddability of the gate-sum graph of two l1-graphs. We have shown that the gate-sum graph of two l1-graphs G1 and G2 is still an l1-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 l1-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, l1-embeddability, gate subgraph, gate-sum, convex cuts

Citation: Wang G, Li C and Wang F (2020) l1-Embeddability Under Gate-Sum Operation of Two l1-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, China

Reviewed by:

Xiaogang Liu, Northwestern Polytechnical University, China
Huiqiu 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