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Folding List of Graphs Obtained from a Given Graph Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2020, Article ID 1316497, 9 pages https://doi.org/10.1155/2020/1316497 Research Article Folding List of Graphs Obtained from a Given Graph E. M. El-Kholy 1 and H. Ahmed2 1Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt 2Department of Mathematics, Faculty of Shoubra Engineering, Banha University, Banha, Egypt Correspondence should be addressed to E. M. El-Kholy; [email protected] Received 6 August 2020; Revised 26 September 2020; Accepted 27 October 2020; Published 12 December 2020 Academic Editor: Irena Lasiecka Copyright © 2020 E. M. El-Kholy and H. Ahmed. ,is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 as a complete bipartite graph from which edges ui; vj; i � 1; ... ; n, have been removed. Let G � (V; E) be a graph, where V is the set of its vertices (6) A simplex graph κ(G) of an undirected graph G is and E is the set of its edges. By a graph, we mean a simple and itself a graph, with a vertex for each clique in G. Two finite connected graph; that is, a graph without multiple vertices of κ(G) are joined by an edge whenever the edges or loops. Let G be a graph, then corresponding two cliques differ in the presence or (1) ,e dual graph G∗ of a graph G is obtained by absence of a single vertex. ,e single vertices are placing a vertex in every face of G and an edge called the zero vertices [6]. joining every two vertices in neighbouring faces [1]. (7) A crossed prism graph for positive even n is a graph C (2) A gear graph, denoted Gn, is a graph obtained by obtained by taking two disjoint cycle graphs n and inserting an extra vertex between each pair of adding edges (vk; v2k+1) and (vk+1; v2k) for adjacent vertices on the perimeter of a wheel k � 1; 3; ... ; (n − 1) [7]. We will denote this graph CP graph Wn. ,us, Gn has 2n + 1 vertices and 3n by n. edges [2]. (8) If two graphs G and H each contain cliques of equal (3) If the edge e joins vertices v and w, then the sub- size, the clique-sum of G and H is formed from their division of e replaces e by a new vertex u and two disjoint union by identifying pairs of vertices in new edges vu and uw [3]. A subdivision of a graph G these two cliques to form a single shared clique, is a graph obtained from G by applying a finite without deleting any of the clique edges [8]. number of subdivisions of edges in succession. (9) Let G1 and G2 be two simple graphs and f: G1 G (4) ,e web graph Wn;r is a graph consisting of r ⟶ 2 a continuous map. ,en, f is called a graph concentric copies of the cycle graphs Cn, with map, if corresponding vertices connected by edges [4]. (i) For each vertex vεV(G1); f(v) is a vertex in n (5) Crown graph on 2 vertices is an undirected graph V(G2). u ; u ; ... ; u with two sets of vertices � 1 2 n� and (ii) For each edge eεE(G1); f(e) is either a vertex or �v1; v2; ... ; vn� and with an edge from ui to vj an edge of the graph G2, i.e., f(e)εV(G2) whenever i ≠ j [5]. ,e crown graph can be viewed or f(e)εE(G2) [9]. 2 International Journal of Mathematics and Mathematical Sciences (10) A graph map f: G1 ⟶ G2 is called a graph folding 3. Graph Folding of the Gear and the if and only if f maps vertices to vertices and edges to Subdivision Graphs edges [10]. (11) A graph G is planar if it can be drawn in the plane in It should be noted that any graph folding of the wheel graph such a way that no two edges meet each other except Wn maps the hup into itself. at a vertex to which they are both incident. Any such drawing is called a plane drawing of G. Any plane Theorem 2. Let Wn be a wheel graph and Gn be the cor- drawing of G divides the plane into regions called responding gear graph. Let f: Wn ⟶ Wn be a graph faces. folding. 3en, the graph map g: Gn ⟶ Gn is defined by (i) g� v ; v � � �v ; v � and gn v ; v o � �v ; v � if and only If the edges and vertices of a face σi of G1 are mapped to i r k s r j s l if fn v ; v o � �v ; v �, where v and v are the extra the edges and vertices of a face σj of G2, then we write i j k l r s v ; v f(σi) � σj. vertices inserting between the adjacent vertices i j and vk; vl, respectively. 2. Graph Folding of the Dual Graph (ii) For the hub v, g(v) � v. Theorem 1. Let G1 and G2 be graphs and f: G1 ⟶ G2 a Proof. Let f: Wn ⟶ Wn be a graph folding, and consider graph folding. Consider the graph map g: G∗ ⟶ G∗ defined 1 2 the edges nvi; vj o, �vk; vl � ∈ E(Wn) such that by fn vi; vj o � �vk; vl �, i.e., f� vi � � �vk � and fn vj o � �vl �. Now, (i) For all v∗ ∈ V(G∗), g(v∗) � v∗ if and only if i 1 i i let vr and vs be the new vertices inserted between the vertices f(σi) � σi, where σi is a face of G1 of the edges nvi; vj o and �vk; vl �, respectively. ,en, we have (ii) If f(σi) � σj and f(σj) � σj where σi and σj are the ∗ ∗ ∗ ∗ four new edges �vi; vr �, nvr; vj o, �vk; vs �, and �vs ; vl � ∈ neighbouring faces, then gn vi ; vj o � gn vk ; vj o ∗ E(G ), but g� v ; v � � �v ; v � and gn v ; v o � �v ; v �, i.e., the where vk is the vertex of the face σk which is n i r k s r j s l neighbouring to σj but not neighbouring to σi map g maps edges of Gn to other edges of Gn. Also, for all ∗ ∗ vi; vk ∈ V(Wn) if f(vi) � vk, then g maps the edge �vi; v � to (iii) If f(σi) � σj and f(σk) � σl, then g(vi ) � (vi ) and ∗ ∗ the edge �vk; v � where v is the hub, and consequently g is a g(vk ) � (vj ) such that each of σi, σk and σj, σl are neighbouring faces graph folding of the gear graph Gn. □ Example 2. Consider the wheel graph W7 and the corre- sponding gear graph G . Let f: W ⟶ W be a graph Proof. Let f: G1 ⟶ G2 be a graph folding. Suppose that σi, 7 7 7 folding defined by f� v ; v � � �v ; v � and f�(v ; v ); (v ; σj, σk, and σl are the faces of the graph G1 such that f(σi) � 2 3 6 5 2 1 2 v3); (v2; v7); (v3; v4); (v3; v7)} �� (v6; v1); (v6; v5); (v6; v7); σj and f(σj) � σj where σi and σj are the neighbouring (v5; v4); (v5; v7)}. ,en, the graph map g: G7 ⟶ G7 defined faces. If σk is neighbouring to σj and not neighbouring to σi, g� v ; v ; v ; v ; v � � �v ; v ; v ; v ; v � ∗ ∗ ∗ by 8 2 9 3 10 13 6 12 5 11 is a graph then there are no edges joining vi and vk in G1 , but each of folding, see Figure 2. In this case, g maps the edges (v1, v8), nv∗; v∗ o nv∗; v∗ o G∗ i j and k j is an edge of 1 . ,us, by the given (v8, v2), (v2, v7), (v2, v9), (v9, v3), (v3, v7), (v3, v10), and (v10, definition, g maps edges to edges. Now, let f(σi) � σj and v4) to the edges (v1, v13), (v13, v6), (v6, v7), (v6, v12), (v12, v5), v v v v v v f(σk) � σl ,en, by the given definition of g, it maps the ( 5, 7), ( 5, 11), and ( 11, 4), respectively. ∗ ∗ ∗ ∗ vertex vi to vl and the vertex vk to vj . Now, since each of the ∗ ∗ Definition 1. For a graph G, if we subdivide each edge once, we faces σi, σk and σj, σl are neighbouring, then each of �vi ; vk � ∗ ∗ ∗ get a new graph Gs, and we will call it the subdivision graph. and nvl ; vj o is an edge of G1 , i.e., the map g maps edges to edges. And, consequently g is a graph folding of the dual Theorem 3. Let G be a graph and Gs the subdivision graph of graph of G1. □ G. Let f: G ⟶ G be a graph folding defined by for all nvi; vj o ∈ E(G), fn vi; vj o � �vk; vl � ∈ E(G). 3en, the graph map g: Gs ⟶ Gs is defined by Example 1.
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