Optimal Embedding of Graphs with Nonconcurrent Longest Paths in Archimedean Tessellations

Optimal graph embeddings represent graphs in a lower dimensional space in a way that preserves the structure and properties of the original graph. Tese techniques have wide applications in felds such as machine learning, data mining, and network analysis. Do we have small (if possible minimal) k -connected graphs with the property that for any j vertices there is a longest path avoiding all of them? Tis question of Zamfrescu (1972) was the frst variant of Gallai’s question (1966): Do all longest paths in a connected graph share a common vertex? Several good examples answering Zamfrescu’s question are known. In 2001, he asked to investigate the family of geometrical lattices with respect to this property. In 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling. Here, we prove that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufciently smaller orders. Te problem of fnding nonconcurrent longest paths in Archimedean tessellations refers to fnding paths in a lattice such that the paths do not overlap or intersect with each other and are as long as possible. Te complexity of embedding graph is still unknown. Tis problem can be challenging because it requires fnding paths that are both long and do not intersect, which can be difcult due to the constraints of the lattice structure.


Introduction
Graph embedding refers to the process of representing vertices (nodes) in a graph in a low-dimensional space, such that the structure and relationships between the nodes are preserved in the embedding. One way to do this is to use a graph layout algorithm, which is a type of algorithm that positions the nodes of a graph in a two-dimensional or three-dimensional space in a way that refects the structure of the graph. Graph layout algorithms can be used to construct embeddings of graphs by placing the nodes in the low-dimensional space according to the relationships between them. Another way to construct embeddings manually is to use domain knowledge to design features that capture the relationships between nodes in the graph and then use these features to embed the nodes in a lowdimensional space. Tis approach requires a deep understanding of the structure and properties of the graph and can be useful for constructing embeddings for specialized applications or for small graphs. For more details and applications of embedding, we refer to [1,2].
A path in a graph G is called a longest path if in G we do not have any other path that is strictly longer. In 1966, Gallai [3] asked: Do all longest paths in a connected graph have a vertex in common? Te frst answer is negative, given by Walther [4] who constructed a 1-connected planar graph of order 25 in which no vertex was in all longest paths. Te best negative answer for Gallai's problem is a nonplanar graph of order 12, independently found by Walther and Voss [5] and Zamfrescu [6], as shown in Figure 1(a). Te optimality of this graph was verifed (using computers) by Brinkmann and Van Cleemput [7]. Nadeem et al. proved the existence of graphs with the empty intersection of their longest cycles as subgraphs of Archimedean lattices [8]. For more results, we refer to [9][10][11][12][13][14][15].
In 1972, Zamfrescu [16] asked an alternative question: Are there k-connected graphs of small (if possible minimal) order such that, for any choice of i vertices, there is a longest path missing all of them? Te same question was raised for planar graphs. Graphs up to connectivity 3 and for i � 1, 2 are known. Te best known planar graph with nonconcurrent (no vertex in common) longest paths is due to Schmitz [17]. It is a 1-connected graph of order 17, as shown in Figure 1(b). Te smallest known path such as a 2-connected graph found by Skupien [18] is of order 26, and the planar one of order 32 is due to Zamfrescu [6].
In 2001, Zamfrescu [19] asked to investigate the equilateral triangular lattice for such graphs, and the frst answer appeared in [20]. Later, Zamfrescu et al. [21] investigated regular square and regular hexagonal lattices, too, see [22] for best known results answering diferent variants of Gallai's question.
A tessellation or tiling of a plane is a pattern resulted from the arrangements of regular polygons without overlapping. Tere are three types of tilings. A regular tiling is a tiling which is formed by using congruent regular polygons all meeting vertex to vertex. An Archimedean or semiregular tessellation is structured by using two or more types of regular polygons ftting together in such a manner that the same polygons in the same cyclic order surround every vertex. By looking at the vertex and counting the sides of all the polygons that meet at the vertex, one is able to name a tiling. Terefore, there are three regular (3 6 ), (4 4 ), and (6 3 ) and eight semiregular , (4.6.12), and (3.12 2 ) tessellations which comprise diferent combinations of (equilateral) triangles, squares, hexagons, octagons, and dodecagons. Figure 2 depicts part of each of the eight semiregular tessellations. Te third type is an example of the so-called nonregular tessellations, which are free of any restriction on the order of the polygons around vertices. Tere are as many such tessellations as you can imagine.
In their 2017 paper, Chang and Yuan [23] demonstrated the existence of 1-connected and 2-connected graphs in semiregular tessellations, showing nonconcurrent longest paths. However, our research advances beyond their fndings by establishing that the graphs presented in [23] could be more optimal. We achieve this by successfully embedding graphs with smaller orders, surpassing the results of Chang and Yuan.

Embeddings of 1-Connected Graphs with Non-concurrent Longest Paths in Archimedean Tessellations
In this section, we describe 1-connected graphs having empty intersection of their longest paths in Archimedean tilings. Graphs presented here are of smaller order than the ones given in [23]. To get our desired results, we use the following lemma of [23]. Let P be a graph homeomorphic to the graph P ′ given in Figure 3. For each edge of P ′ , the labels in Figure 3 indicate the number of vertices of degrees one and two in the corresponding path of P.
Our frst result is as follows: Tere exists a graph of order 32 with nonconcurrent longest paths embeddable in (3 4    Proof. Te constraints given in Lemma 1 are also verifed for x � 4, y � 0, z � 2, w � 7, v � 9, and u � 0, and the resulting  Proof. Here, we use another particular case of Lemma 1. Te graph P with the desired property is obtained by using x � 9, y � 3, z � 6, w � 16, v � 19, and u � 0 and consists of 114 vertices. An embedding of P in (4.8 2 ) is illustrated in Figure 9. □ Theorem 6. In (4.6.12), we have a graph with nonconcurrent longest paths consisting of 112 vertices.

Complexity 3
Proof. An embedding of the desired graph is presented in Figure 10, which is obtained by setting x � 8, y � 5, z � 5, w � 16, v � 19, and u � 0 in Lemma 1. □ Theorem 7. In the Archimedean tiling (3.12 2 ), we have a graph of order 152 whose longest paths share no vertex in common.

Embeddings of 2-Connected Graphs with Nonconcurrent Longest Paths in Archimedean Tessellations
Tis section is devoted to 2-connected graphs in which longest paths share no vertex and are embeddable in Archimedean tilings. We start with the following lemma of [21]. Let Q be a graph homeomorphic to the graph Q ′ , as shown in Figure 12, where the variables x, y, z, t, and s represent the number of vertices of degree 2 on the corresponding paths to edges mentioned on the respective fgures as well.

Complexity 5
Te graph Q ′ of Figure 12 is not embeddable in the cubic regular tilings (4.8 2 ), (4.6.12), and (3.12 2 ) as it contains vertices of degree 4. To achieve results in the mentioned tilings, we considered the following lemma of [23].
Let R be a graph homeomorphic to the graph R ′ of Figure 18(a), which contains three isomorphic subgraphs homeomorphic to the graph P ′ given in Figure 18(b), where x, y, z, w, v, and u represent the number of vertices of degree 2 on the corresponding paths. [23]. Te longest paths of R are nonconcurrent if
Proof. If we take x � 16, y � 7, z � 9, w � 3, v � 14, and u � 0 in Lemma 14, the resulting graph R of order 367 has the desired property. An embedding of R is presented in Figure 19. Proof. An embedding of the graph R of order 424 obtained under the conditions of Lemma 14 by setting x � 18, y � 5, z � 13, w � 6, v � 18, and u � 0 is shown in Figure 20.

Conclusion
In 2001, Zamfrescu [19] raised a question: Do we have small (if possible minimal) k-connected graphs with the property that for any j vertices there is a longest path avoiding all of them in lattices? To answer this question, in 2017, Chang and Yuan proved the existence of such graphs in Archimedean tiling [23]. Here, we proved that the graphs presented by Chang and Yuan are not optimal by constructing such graphs of sufciently smaller orders. For comparison, see Table 1 given below. We conclude this paper with the following problems: Open Problem 18. Find embeddings of smaller order than those presented in Teorems 2-7 and 9-17. Moreover, we fnd lattices of smaller order than those of Sections 2 and 3, admitting the desired embeddings.
Open Problem 19. Find embeddings of 2-connected graphs with the property that for any j vertices there is a longest path/cycle avoiding all of them.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.