Abstract
In this paper, we examine the relation between graph folding of a given graph and foldings of new graphs obtained from this graph by some techniques like dual, gear, subdivision, web, crown, simplex, crossed prism, and clique-sum graphs. In each case, we obtained the necessary and sufficient conditions, if exist, for these new graphs to be folded.
1. Introduction
Let be a graph, where V is the set of its vertices and E is the set of its edges. By a graph, we mean a simple and finite connected graph; that is, a graph without multiple edges or loops. Let G be a graph, then(1)The dual graph of a graph G is obtained by placing a vertex in every face of G and an edge joining every two vertices in neighbouring faces [1].(2)A gear graph, denoted , is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph . Thus, has vertices and edges [2].(3)If the edge e joins vertices and , then the subdivision of e replaces e by a new vertex u and two new edges and [3]. A subdivision of a graph G is a graph obtained from G by applying a finite number of subdivisions of edges in succession.(4)The web graph is a graph consisting of r concentric copies of the cycle graphs , with corresponding vertices connected by edges [4].(5)Crown graph on vertices is an undirected graph with two sets of vertices and and with an edge from whenever [5]. The crown graph can be viewed as a complete bipartite graph from which edges , have been removed.(6)A simplex graph of an undirected graph G is itself a graph, with a vertex for each clique in G. Two vertices of are joined by an edge whenever the corresponding two cliques differ in the presence or absence of a single vertex. The single vertices are called the zero vertices [6].(7)A crossed prism graph for positive even n is a graph obtained by taking two disjoint cycle graphs and adding edges and for [7]. We will denote this graph by .(8)If two graphs G and H each contain cliques of equal size, the clique-sum of G and H is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, without deleting any of the clique edges [8].(9)Let be two simple graphs and a continuous map. Then, f is called a graph map, if(i)For each vertex is a vertex in .(ii)For each edge is either a vertex or an edge of the graph , i.e., [9].(10)A graph map is called a graph folding if and only if f maps vertices to vertices and edges to edges [10].(11)A graph is planar if it can be drawn in the plane in such a way that no two edges meet each other except at a vertex to which they are both incident. Any such drawing is called a plane drawing of . Any plane drawing of divides the plane into regions called faces.
If the edges and vertices of a face of are mapped to the edges and vertices of a face of , then we write .
2. Graph Folding of the Dual Graph
Theorem 1. Let be graphs and a graph folding. Consider the graph map defined by(i)For all , = if and only if , where is a face of (ii)If and where and are the neighbouring faces, then = where is the vertex of the face which is neighbouring to but not neighbouring to (iii)If and , then and such that each of , and , are neighbouring faces
Proof. Let be a graph folding. Suppose that , , , and are the faces of the graph such that and where and are the neighbouring faces. If is neighbouring to and not neighbouring to , then there are no edges joining and in , but each of and is an edge of . Thus, by the given definition, maps edges to edges. Now, let and Then, by the given definition of , it maps the vertex to and the vertex to . Now, since each of the faces , and , are neighbouring, then each of and is an edge of , i.e., the map maps edges to edges. And, consequently is a graph folding of the dual graph of .
Example 1. Consider the graphs shown in Figure 1(a). Let be a graph folding defined by and , i.e., . The graph map defined is a graph folding, see Figure 1(b). The omitted vertices and edges, or faces, are mapped to themselves through this paper.
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3. Graph Folding of the Gear and the Subdivision Graphs
It should be noted that any graph folding of the wheel graph maps the hup into itself.
Theorem 2. Let be a wheel graph and be the corresponding gear graph. Let be a graph folding. Then, the graph map is defined by(i) = and = if and only if = , where are the extra vertices inserting between the adjacent vertices and , respectively.(ii)For the hub , = .
Proof. Let be a graph folding, and consider the edges , such that = , i.e., = and = . Now, let be the new vertices inserted between the vertices of the edges and , respectively. Then, we have four new edges , , , and , but = and = , i.e., the map maps edges of to other edges of . Also, for all if = , then maps the edge to the edge where is the hub, and consequently is a graph folding of the gear graph .
Example 2. Consider the wheel graph and the corresponding gear graph . Let be a graph folding defined by and . Then, the graph map defined by is a graph folding, see Figure 2. In this case, maps the edges (, ), (, ), (, ), (, ), (, ), (, ), (, ), and (, ) to the edges (, ), (, ), (, ), (, ), (, ), (, ), (, ), and (, ), respectively.
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Definition 1. For a graph G, if we subdivide each edge once, we get a new graph , and we will call it the subdivision graph.
Theorem 3. Let be a graph and the subdivision graph of . Let be a graph folding defined by for all , = . Then, the graph map is defined by(i)Mapping the edges and to themselves if and only if f maps the edge to itself(ii) = and = where are the new vertices replaced for the edges and , respectively, is a graph folding
Proof. Suppose that is a graph folding such that = where . Now, replace the edge by the new edges and the edge by the new edges . Since = and = , then maps edges to edges of .
Now, let be an edge of such that . The subdivision of replaces by a new vertex and two new edges and . Since and , then maps edges to edges of . Thus, is a graph folding.
Example 3. Consider the graph G and its subdivision Gs shown in Figure 3. Let be a graph folding defined by and . Then, the graph map defined by and is a graph folding.
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4. Graph Folding of the Web and Crown Graphs
Theorem 4. Let be a cycle graph, where , n is even, and be the web graph where . Let be a graph folding defined by for all , . Then, the graph map is defined by(i)For all , = = (ii)For all , , then =
Proof. Let be a cycle graph with even vertices and a graph folding defined by for all , . Consider the vertices such that f maps the edge to the edge . For the vertices , = = , and hence is a graph folding. If s = 2, then = and = , i.e., maps edges to edges. The same procedure can be done if . Thus, is a graph folding. For illustration, see Figure 4.
Example 4. Consider the cycle graph . Let be a graph folding defined by . The graph map defined by and is a graph folding, see Figure 5.
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Theorem 5. Any crown graph G of 2n vertices can be folded to an edge.
Proof. Let G be a crown graph on 2n vertices, and let = and = be the two sets of vertices of G. Now, a graph folding can be defined by mapping all the vertices of to one vertex of , say , and all the vertices of to one vertex of , say . Thus, the image is the edge . Thus, is a graph folding. For illustration, see Figure 6.
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5. Graph Folding of the Simplex and Crossed Prism Graphs
Theorem 6. Let be a graph and a graph folding. Then, the graph map is defined by(i)For a zero vertex , if = , then = where , .(ii)If and are cliques of such that = , then = and = where are the new vertices of the cliques and , respectively.(iii)If and are cliques of such that = , then = , = 1, 2, 3, where are the new vertices of the two cliques and and are the new vertices of the edges of and , respectively.And so on.
Proof. Let be a graph and a graph folding(i)Consider the vertices , such that = . Let be a zero vertex of , and then by the given definition of , it maps the vertex onto itself. Then, we get new edges , but = , i.e., maps edges to edges of .(ii)Consider the cliques and of such that = . Let and be the new vertices of the two cliques, respectively; then we have new four edges, , , and but = and = , i.e., maps edges to edges.(iii)Finally, let and be cliques of such that = . And, considering the new vertices and of the two cliques and , respectively. Then, we have new edges , where and are the new vertices of the edges of the cliques and , respectively. Then, the map maps the new edges of the boundary of to the new edges of the boundary of according to the rule (ii), and = where . For illustration, see Figure 7. Hence, is a graph folding of the simplex graph .
Example 5. Let be the graph shown in Figure 8(a) and the graph folding defined by and . Then, the graph map defined by and is a graph folding, see Figure 8(b).
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Theorem 7. Let be a cycle graph of even vertices and a graph folding. Consider the graph map such that for all , :(i)If = whenever = , then is a graph folding(ii)If = whenever = , then is not a graph folding
Proof. (i)Let be a graph folding, and consider the edges such that = . Now, = for all . Also, = , i.e., maps the edge to the edge . Consider one of the new edges of , e.g., . Then, = , i.e., maps edges to edges, and hence is a graph folding.(ii)Now, let = , and we know that for all , = . Also, for the edge = , but if we consider one of the new edges of , e.g., , then = which is not an edge of , and hence is not a graph folding.
Example 6. Consider the cycle graph , and let be a graph folding defined by and . The graph map defined by is not a graph folding since , see Figure 9(a). While if is a graph folding defined by , then the graph map defined by and is a graph folding, see Figure 9(b).
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6. Graph Folding of the Clique-Sum Graph
We will denote the clique-sum of the two graphs G and H by G cli H.
Definition 2. Let , , , and be graphs. Let and be graph maps. Then, we can define a map from the clique-sum of and to the clique-sum of and denoted by as follows:
for all edges of the shared cliques.
This map is called the clique-sum map of the maps and .
Theorem 8. Let , , , and be graphs. Let and be graph foldings. Then, the clique-sum map is a graph folding.
Proof. Suppose and are graph foldings. Now, let , and then either or . In these two cases and since each of and is a graph folding, then . Thus, maps edges to edges, and hence the clique-sum map is a graph folding.
Example 7. Consider the two graphs and shown in Figure 10(a). Let be a graph folding defined by and . Then, the clique-sum map defined by and is a graph folding, see Figure 10(b).
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7. Conclusion
We obtained the necessary and sufficient conditions, if exist, for folding new graphs obtained from a given graph by some techniques like dual, gear, subdivision, web, crown, simplex, crossed prism, and clique-sum graphs. We can examine the relation between folding a given pair of graphs and folding of a new graph generated from these given pair of graphs by some operations like join, Cartesian product, normal product, and tensor product. Also, we can lift the definition of folding from graphs to digraphs which has close connections to important industrial applications.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.