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 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 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 g� v , v � � �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 f� v , v � � �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 �vi, vj �, �vk, vl � ∈ E(Wn) such that by f� vi, vj � � �vk, vl �, i.e., f� vi � � �vk � and f� vj � � �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 �vi, vj � 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 �, �vr, vj �, �vk, vs �, and �vs , vl � ∈ neighbouring faces, then g� vi , vj � � g� vk , vj � ∗ E(G ), but g� v , v � � �v , v � and g� v , v � � �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), �v∗, v∗ � �v∗, v∗ � 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 �vl , vj � 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 �vi, vj � ∈ E(G), f� vi, vj � � �vk, vl � ∈ E(G). 3en, the graph map g: Gs ⟶ Gs is defined by Example 1. Consider the graphs G1 and G2 shown in Figure 1(a). Let f: G1 ⟶ G2 be a graph folding defined by (i) Mapping the edges vu and uw to themselves if and f(v6) � (v1) and f�(v6, v2), (v6, v3), (v6, v4), (v6, v5)} � only if f maps the edge vw to itself �(v1, v2), (v1, v3), (v1, v4), (v1, v5)�, i.e., f�σ5, σ6, σ7, σ8} � (ii) g� vi, ur � � �vk, us � and g� ur, vj � � �us, vl � where �σ1, σ2, σ3 , σ4�. ,e graph map g: G1 ∗ ⟶ G2 ∗ defined ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ur and us are the new vertices replaced for the edges g� v5 , v6 , v7 , v8 � � �v3 , v4 , v1 , v2 } and �(v5 , v1 ), (v5 , v6 ), �v , v � �v , v � ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ i j and k l , respectively, is a graph folding (v5 , v7 ), (v6 , v2 ), (v6 , v8 ),(v7 ,v3 ), (v7 , v8 ), (v8 , v4 )} �� (v3 , ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ v1 ), (v3 , v4 ), (v3 , v1 ), (v4 , v2 ), (v4 , v2 ), (v1 , v3 ), (v1 , v2 )} is a graph folding, see Figure 1(b). ,e omitted vertices Proof. Suppose that f: G ⟶ G is a graph folding such that and edges, or faces, are mapped to themselves through this f� vi, vj � � �vk, vl � where �vi, vj ��vk, vl � ∈ E(G). Now, re- paper. place the edge �vi, vj � by the new edges International Journal of Mathematics and Mathematical Sciences 3

v1 σ σ3 1 σ σ 4 2 v1 σ 3 σ1

σ2 v v σ4 4 5 v2 f

v3 v5 σ7 v σ5 4 v2 v3 σ8 v6 σ6 f (G1) = G2 G1

(a)

∗ ∗ v4 v2 ∗ v4 ∗ v2 ∗ v ∗ v1 3 g

∗ ∗ v3 v1 ∗ ( ∗) = ∗ v8 ∗ g G1 G2 v6

v ∗ ∗ 7 ∗ v5 G1

(b)

Figure 1: Graph folding of a graph and its dual graph.

C �vi, ur ��ur, vj � ∈ E(Gs) and the edge �vk, vl � by the new Proof. Let n be a with even vertices and f: Cn ⟶ Cn a graph folding defined by for all edges �vk, us ��us, vl � ∈ E(Gs). Since g� vi, ur � � �vk, us � and v ∈ V(C ), f(v ) � v ∈ V(C ). Consider the vertices g� u , v � � �u , v �, then g maps edges to edges of G . i n i j n r j s l s v , v , v , v ∈ V(C ) such that f maps the edge �v , v � to the Now, let e � vw be an edge of G such that f(vw) � vw. i j k l n i k edge �v , v �. For the vertices v ∈ V(C ), g(v ) � f(v ) � v , ,e subdivision of e replaces e by a new vertex u and two j l i n i i j and hence g is a graph folding. If s � 2, then new edges vu and uw. Since g(vu) � vu and g(uw) � uw, g� v , v � � �v , v � and g� v , v � � �v , v �, i.e., then g maps edges to edges of G . ,us, g is a graph i i+n j j+n i+n k+n j+n l+n s g maps edges to edges. ,e same procedure can be done if folding. □ s � 3, 4, ... , r. ,us, g is a graph folding. For illustration, see Figure 4. □ Example 3. Consider the graph G and its subdivision Gs f: G G shown in Figure 3. Let ⟶ be a graph folding Example 4. Consider the cycle graph C4. Let f: C4 ⟶ C4 f(v ) � v f� e , e � � �e , e � defined by 1 3 and 1 2 4 3 . ,en, the be a graph folding defined by f(v2) � (v4) and g: G G g(v , u , u ) � graph map s ⟶ s defined by 1 1 2 f�(v2, v1), (v2, v3)� �� (v4, v1), (v4, v3)�. ,e graph map (v , u , u ) g� e′, e′, e′, e′ � � �e′, e′, e′, e′ � 3 4 3 and 1 2 7 8 4 3 6 5 is a graph g: W4,3 ⟶ W4,3 defined by g� v2, v6, v10 � � �v4, v8, v12 � and folding. g�(v2,v1),(v2,v3),(v2, v6), (v6, v5), (v6, v7), (v6, v10), (v10, v9), (v10, v11)} �� (v4, v1), (v4, v3), (v4, v8), (v8, v5), (v8, v7), (v8, v ), (v , v ), (v , v )} 4. Graph Folding of the Web and Crown Graphs 12 12 9 12 11 is a graph folding, see Figure 5. Theorem 5. Any crown graph G of 2n vertices can be folded Theorem 4. Let Cn be a cycle graph, where V(Cn) � to an edge. �v1, ... , vn �, n is even, and Wn,r be the web graph where V(Wn,r) � �v1, ... , vn , vn+1 , ... , v2n , v2n+1, ... , v3n , ... , Proof. Let G be a crown graph on 2n vertices, and let v(r−1)n+1 , ... , vrn}. Let f: Cn ⟶ Cn be a graph folding A � �u , u , ... , u � and B � �v , v , ... , v � be the two sets of defined by for all vi ∈ V(Cn), f(vi) � vj ∈ V(Cn). 3en, the 1 2 n 1 2 n vertices of G. Now, a graph folding f: G ⟶ G can be graph map g: Wn,r ⟶ Wn,r is defined by defined by mapping all the vertices of A to one vertex of A, (i) For all v ∈ V(C ), g(v ) � f(v ) � v i n i i j say ui, and all the vertices of B to one vertex of B, say vj. ,us, (ii) For all vi+(s−1)n ∈ V(Wn,r) , s � 2, ... , r, then the image f(G) is the edge �ui, vj � ∈ E(G). ,us, f is a g(vi+(s−1)n) � vj+(s−1)n graph folding. For illustration, see Figure 6. □ 4 International Journal of Mathematics and Mathematical Sciences

v6 v1 v 6 v1

f v5 v2 v7 v5 v7

v4 v4 v3 W7 f (W7)

(a)

v6 v1 v13 v6 v1 v13 v12 v8 v12 g v5 v2 v7 v5 v7 v11 v9 v11

v10 v4 v4 v3 G7 g (G7)

(b) Figure 2: Graph folding of the wheel graph W7 and the gear graph G7.

v1 v4 e1 v4

e5 f e2 e4 e5 e4

e3 e3 v2 v3 v2 v3 G f (G) (a) ′ ′ e1 u1 e2 v1 v4 v4 ′ ′ ′ e10 ′ e8 e10 ′ e3 e3 g u5 u5 u2 u4 u4 ′ ′ e9 ′ e9 ′ e7 ′ e4 e4 ′ ′ ′ ′ e6 u3 e5 e6 u3 e5 v2 v3 v2 v3 ( ) g Gs Gs (b)

Figure 3: Graph folding of a graph and its subdivision graph.

5. Graph Folding of the Simplex and Crossed g� vr, vj � � �vs, vl � where vr and vs are the new ver-

Prism Graphs tices of the cliques �vi, vj � and �vk, vl �, respectively. � �v , v , v � � �v , v , v � G Theorem 6. Let G be a graph and f: G ⟶ G a graph (iii) If σ i j k and δ l m n are cliques of folding. 3en, the graph map g: k(G) ⟶ k(G) is defined by such that f� v , v , v � � �v , v , v �, then g� u , v � i j k l m n rμ (i) For a zero vertex v ∈ V(k(G)), if f(vi) � vk, then � �w , v �, μ � 1, 2, 3, where u and w are the new sμ g� vi, v � � �vk, v � where vi, vk ∈ V(G). vertices of the two cliques σ and δ and vr and vs are (ii) If �vi, vj � and �vk, vl � are cliques of G such that μ μ the new vertices of the edges of σ and δ, respectively. f� vi, vj � � �vk, vl �, then g� vi, vr � � �vk, vs � and And so on. International Journal of Mathematics and Mathematical Sciences 5

v2

v1 vn+2 v3 vn+1 vn+3 v2n+2 v2n+1 v(r–1)n+2 v2n+3 v(r–1)n+1 v(r–1)n+3 v(r–1)n+4 vrn v2n+4

vn+4 v3n v4 v2n

vn Figure 4: ,e web graph.

v4 v1 v4 v1 f

v3 v2 v3 C4 f (C4)

(a)

v4 v1

v8 v5 v4 v1 v12 v8 v v9 g 5 v12 v9 v11 v10 v6 v11 v7 v7 v3 v2 W4,3 v3 g (W4,3)

(b) Figure 5: Graph folding of the cycle graph C4 and the web graph W4,3.

Proof. Let G be a graph and f: G ⟶ G a graph folding the new edges of the boundary of σ to the new edges of the boundary of δ according to the rule (ii), (i) Consider the vertices vi, vk ∈ V(G) such and g�u, vr � � �w, vs � where �u, vr ��w, vs � ∈ thatf(vi ) � vk. Let v be a zero vertex of V(k(G)), μ μ μ μ and then by the given definition of g, it maps the E(k(G)). For illustration, see Figure 7. Hence, g is a vertex v onto itself. ,en, we get new edges graph folding of the simplex graph k(G). □ �vi, v �,�vk, v � ∈ E(k(G)) but g� vi, v � � �vk, v �, i.e., g maps edges to edges of (k(G)). Example 5. Let G be the graph shown in Figure 8(a) and (ii) Consider the cliques �v , v � and �v , v � of G such i j k l f: G ⟶ G the graph folding defined by f� v3, v6, v8 � � f v , v � v , v v v that �i j � �k l �. Let r and s be the new �v1, v4, v7 � and f�(v5, v6), (v2, v3), (v3, v6), (v6, v8), (v3, vertices of the two cliques, respectively; then we v8)} �� (v5, v4), (v2, v1), (v1, v4), (v4, v7), (v1, v7)�. ,en, the g: k(G) k(G) g� v , v , v , have new four edges�vi, vr �, �vr, vj �, �vk, vs �, and graph map ⟶ defined by 3 6 8 u7, u8, u9, u10, u11, w2} � �v1, v4, v7, u4, u5, u3, u2, u1, w1� and �vs, vl � ∈ E(k(G)) but g� vi, vr � � �vk, vs � and g�(v3,V), (v6,V), (v8,V), (v5, u8), (u8, v6), (v2, u7), (u7, g� vr, vj � � �vs, vl �, i.e., g maps edges to edges. v3), (v3, u9), (u9, v6), (v6, u10), (u10, v8), (v3, u11), (u11, v8), (iii) Finally, let σ � �vi, vj, vk � and δ � �vl, vm, vn � be (w2, u9), (w2, u10), (w2, u11)} �� (v1,V), (v4,V), (v7,V), (v , u ), (u , v ), (v , u ), (u , v ), (v , u ), (u , v ), (v , u ), cliques of G such that f� vi, vj, vk � � �vl, vm, vn �. 5 5 5 4 2 4 4 1 1 3 3 4 4 2 (u , v ), (v , u ), (u , v ), (w , u ), (w , u ), (w , u )} And, considering the new vertices u and w of the 2 7 1 1 1 7 1 3 1 2 1 1 is a two cliques σ and δ, respectively. ,en, we have new graph folding, see Figure 8(b).

edges �u, vr ��w, vs � ∈ E(k(G))μ � 1, 2, 3, where μ μ Theorem 7. Let Cn be a cycle graph of even vertices and v and v are the new vertices of the edges of the f: C C rμ sμ n ⟶ n a graph folding. Consider the graph map cliques σ and δ, respectively. ,en, the map g maps g: CPn ⟶ CPn such that for all e ∈ E(Cn), g{} e � f{} e : 6 International Journal of Mathematics and Mathematical Sciences

u1 v3 u1 u3 u2 u1 u1 f f1 u 1 3 v2 or v2 v2 v3 v2 v1 ( ) v1 u2 f1 (G1) f1 G1 G1 G1

(a)

u4 v3 u1 u4 u3 u2 u1 v3

v2 u1 f2 or f2

u1 v3 u3 v4 v4 v3 v2 v1 f2 (G2) f2 (G2) v1 u2 G2 G2

(b) Figure 6: Folding crown graphs with six and eight vertices to an edge. (a) f1 � �u1, u2, u3 � � u1, f1 � �v1, v2, v3 � � v2 and (b) f2 � �u1, u2, u3, u4 � � u1, f2 � �v1, v2, v3, v4 � � v3.

vl vi

vk v vn vm σ j δ

(a)

vl vi

vr vs 3 v 3 vs r1 1 u w

v v vn k v j vs vm r2 2 k (σ) k (δ)

(b)

Figure 7: ,e simplex graph.

(i) If g� v2k, v2k+1 � � �v2k+i, v2k+i+1 � whenever f� vk, vk+1 � g� vk+1, v2k � � �vk+i+1, v2k+i � ∈ E(CPn), i.e., g maps edges to edges, and hence g is a graph � �vk+i, vk+i+1 �, then g is a graph folding folding. (ii) If g� v2k, v2k+1 � � �v2k+i, v2k+1 � whenever f� vk, vk+1 � (ii) Now, let f� vk, vk+1 � � �vk+i, vk+1 �, and we know that � �vk+i, vk+1 �, then g is not a graph folding for all e ∈ E(Cn), g{} e � f{} e . Also, for the edge �v2k, v2k+1 � ∈ E(CPn)g� v2k, v2k+1 � � �v2k+i, v2k+1 � E(CP ) Proof (i) Let f: C ⟶ C be a graph folding, and con- ∈ n , but if we consider one of the new edges of n n CP �v , v � g� v , v � � �v , v � sider the edges �v , v ��v , v � ∈ E(C ) n, e.g., k 2k+1 , then k 2k+1 k+i 2k+1 k k+1 k+i k+i+1 n CP g such thatf� v , v � � �v , v �. Now, which is not an edge of n, and hence is not a k k+1 k+i k+i+1 graph folding. □ g{} e � f{} e for all e ∈ E(Cn). Also, g� v2k, v2k+1 � � �v2k+i, v2k+i+1 �, i.e., g maps the edge �v2k, v2k+1 � ∈ E(CPn) to the edge Example 6. Consider the cycle graph C4, and let �v2k+i, v2k+i+1 � ∈ E(CPn). Consider one of the f1: C4 ⟶ C4 be a graph folding defined by f1(v1) � (v3) new edges of CPn, e.g., �vk+1, v2k �. ,en, International Journal of Mathematics and Mathematical Sciences 7

v7 v7

v1 v4 v1 v4 f v2 v5 v2 v5 v3 v6 f (G) v8 G (a)

v7

u1 w1 u2 u3 v4 v7 v1 u1 w u2 v 1 u u4 u5 3 v4 g v1 v5 v2 v u6 u u4 5 u7 u8 u9 v2 v3 u6 v5 v6 w2 u u11 10 g (k (G)) v8 k (G) (b)

Figure 8: Graph folding of the simplex graph.

v1 v2 v2 v1 v2 v2 u2 u1 u2 f1 g1 and u4 u3 u4 u3 v4 v3 v3 v4 v3 C4 ( ) v4 f1 C4 v4 v3 g1 (CP4) CP4

(a)

v1 v2 v1 v2 u1 u2 u4 u3 f2 g2 v4 v3 and v3 f (C ) v4 2 4 u4 u3 g2 (CP4) v4 v3 v4 v3 CP4

(b)

Figure 9: ,e crossed prism graph may or may not be folded.

and f1�(v1, v2), (v1, v4)� �� (v3, v2)(v3, v4)�. ,e graph map by f2(v1, v2) � (v3, v4) and f2�(v1, v2), (v1, v4), g1: CP4 ⟶ CP4 defined by g1�v1, u1 � � �v3, u3 � is not a (v2, v3)} �� (v3, v4), (v3, v4), (v4, v3)�, then the graph map graph folding since g1(v1, u2) � (v3, u2) ∉ E(CP4), see g2: CP4 ⟶ CP4 defined by g2�v1, v2, u1, u2 � � �v3, v4, Figure 9(a). While if f2: C4 ⟶ C4 is a graph folding defined u3, u4} and g2�(v1, v2), (v1, v4), (v2, v3), (u1, u2), (u1, u4), 8 International Journal of Mathematics and Mathematical Sciences

e4 v1 v4 e7 e4 e5 v1 e1 v4 v2 f e5 e3 v2 e1 e6 e3 e8 e2 e2 v5 e6 e9 G1 v3 f (G1) v3

v1 e10 v8

v1 e1 e13 e11 e14 e1 v2 e12 g e3 v7 e3 v2 e17 e15 e2 e16 e2 e18 v3 v6 v3 G2 g (G2)

(a)

v1 e10 v8 e4 v4 e7 e13 e5 e1 e4 v1 e14 fclig v2 e11 v4 e12 e3 v7 e5 v2 e1 e e e17 8 6 e6 e3 v5 e2 e15 e16 e2 e9 e18 v3 v6 ( cli ) ( cli ) v3 G1cliG2 f g G1 G2

(b)

Figure 10: Graph folding of the clique-sum graph.

(u2, u3), (v1, u2), (v2, u1)} �� (v3, v4), (v3, v4), (v4, v3), (u3, (fclig)(e) ∈ (G3cli G4). ,us, (fcli g) maps edges to edges, u4), (u3, u4), (u4, u3), (v3, u4), (v4, u3)} is a graph folding, see and hence the clique-sum map is a graph folding. □ Figure 9(b).

Example 7. Consider the two graphs G1 and G2 shown in 6. Graph Folding of the Clique-Sum Graph Figure 10(a). Let f: G1 ⟶ G1 be a graph folding defined by f� v5 � � �v4 �, f� e7, e8, e9 � � �e4, e5, e6 �, g� v6, v7, v8 � � �v1, v2, We will denote the clique-sum of the two graphs G and H by v3} and g� e17, e18, e12, e16, e10, e11, e13, e14, e15� � �e1, e3, e3, G cli H. e1, e3, e2, e2, e1, e2}. ,en, the clique-sum map fcli g : G1cli G2 ⟶ G1cliG2 defined by (fcli g)�v5, v6, v7, v8 � � �v4, v , v , v } and (fcli g)�e , e , e , e , e , e , e , e , e , e , Definition 2. Let G1, G2, G3 , and G4 be graphs. Let 1 2 3 7 8 9 17 18 12 16 10 11 13 e , e } � �e , e , e , e , e , e , e , e , e , e , e , e � is a graph f: G1 ⟶ G3 and g: G2 ⟶ G4 be graph maps. ,en, we 14 15 4 5 6 1 3 3 1 3 2 2 1 2 folding, see Figure 10(b). can define a map from the clique-sum of G1 and G2 to the clique-sum of G3 and G4 denoted by fcli g : G1 cli G2 ⟶ G3 cli G4 as follows: 7. Conclusion ⎧⎪ f(e), e ∈ G , ⎨ 1 We obtained the necessary and sufficient conditions, if (fcli g)(e) � f(e) � g(e) � e, for all edges of the ⎩⎪ exist, for folding new graphs obtained from a given graph g(e), e ∈ G2, shared cliques. by some techniques like dual, gear, subdivision, web, ,is map is called the clique-sum map of the maps f and crown, simplex, crossed prism, and clique-sum graphs. g. 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, Car- Theorem 8. Let G1, G2, G3 , and G4 be graphs. Let tesian product, normal product, and tensor product. Also, f: G1 ⟶ G3 and g: G2 ⟶ G4 be graph foldings. 3en, the we can lift the definition of folding from graphs to di- clique-sum map fcli g : G1cli G2 ⟶ G3cliG4 is a graph graphs which has close connections to important in- folding. dustrial applications.

Proof. Suppose f and g are graph foldings. Now, let Data Availability e ∈ G1cli G2, and then either e ∈ G1 or e ∈ G2. In these two cases and since each of f and g is a graph folding, then No data were used to support this study. International Journal of Mathematics and Mathematical Sciences 9

Conflicts of Interest ,e authors declare that they have no conflicts of interest regarding the publication of this paper. References [1] R. Balakrishnan and K. Ranganathan, A Textbook of Graph 3eory, Springer Science & Business Media, Berlin, Germany, 2012. [2] A. Brandstadt and J. P. Spinrad, Graph Classes: A Survey, vol. 3, SIAM, Philadelphia, PA, USA, 1999. [3] R. J. Trudeau, Introduction to Graph 3eory, Courier Cor- poration, Chelmsford, MA, USA, 2013. [4] J. A. Gallian, “A dynamic survey of graph labeling,” 3e Electronic Journal of Combinatorics, vol. 19, 2000. [5] A. E. Brouwer and W. H. Haemers, “Distance-regular graphs,” in Universitext, Spectra of Graphs Springer, New York, NY, USA, 2012. [6] H.-J. Bandelt and M. Van De Vel, “Embedding topological median algebras in products of dendrons,” Proceedings of the London Mathematical Society, vol. 58, no. 3, pp. 439–453, 1989. [7] R. C. Read and R. J. Wilson, An Atlas of Graphs, Vol. 21, Clarendon Press, Oxford, UK, 1998. [8] P. D. Seymour and R. W. Weaver, “A generalization of chordal graphs,” Journal of Graph 3eory, vol. 8, no. 2, pp. 241–251, 1984. [9] P. Erd¨os,“ and probability,” Canadian Journal of Mathematics, vol. 11, pp. 34–38, 1959. [10] E. El-Kholy and A. El-Esawy, “Graph folding of some special graphs,” Journal of Mathematics and Statistics, vol. 1, no. 1, pp. 66–70, 2005.