End-regular and End-orthodox generalized lexicographic products of bipartite graphs

Rui Gu; Hailong Hou

doi:10.1515/math-2016-0021

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End-regular and End-orthodox generalized lexicographic products of bipartite graphs

Rui Gu and Hailong Hou

Published/Copyright: April 6, 2016

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$Open Mathematics$

From the journal Open Mathematics Volume 14 Issue 1

Abstract

A graph X is said to be End-regular (End-orthodox) if its endomorphism monoid End(X) is a regular (orthodox) semigroup. In this paper, we determine the End-regular and the End-orthodox generalized lexicographic products of bipartite graphs.

Keywords: Endomorphism; Monoid; Generalized lexicographic product; Bipartite graph

MSC 2010: 05C25

1 Introduction and preliminary concepts

Endomorphism monoids of graphs are generalizations of automorphism groups of graphs. In recent years much attention has been paid to endomorphism monoids of graphs and many interesting results concerning graphs and their endomorphism monoids have been obtained (see [1, 2, 5, 11–14] and references therein). The aim of this research is to develop further relations between graph theory and algebraic theory of semigroups and to apply the theory of semigroups to graph theory. Petrich and Reilly [18] pointed out that, in the great range of special classes of semigroups, regular semigroups take a central position from the point of view of richness of their structural regularity. So it is natural to ask: which graphs have regular endomorphism monoids? This question was posed by L.Marki [17] as an open problem. However, it seems difficult to obtain a general answer to this question. So the strategy for solving this question is finding various kinds of conditions of regularity for various kinds of graphs. In [19], connected bipartite End-regular graphs were explicitly found. A family of circulant complete graphs K(n, 3) whose endomorphism monoids are regular was described in [6]. Hou, Luo and Cheng [8] explored the endomorphism monoid of P_n, the complement of a path P_n with n vertices. It was shown that P_n is End-orthodox. The split graphs with regular endomorphism monoids were characterized in [14]. The joins of split graphs with regular endomorphism monoids were studied in [7]. The joins of bipartite graphs with regular endomorphism monoids were studied in [9]. In this paper, we will determine the End-regular and the End-orthodox generalized lexicographic products of connected bipartite graphs.

The graphs X considered in this paper are finite undirected graphs without loops and multiple edges. The vertex set of X is denoted by V (X) and the edge set of X is denoted by E(X). If two vertices x₁ and x₂ are adjacent in graph X, the edge connecting x₁ and x₂ is denoted by {x₁, x₂} and write {x₁, x₂} ∊ E(X). A graph X is called bipartite if X has no odd cycle. It is known that, if a graph X is bipartite, then its vertex set V(X) can be partitioned into two disjoint non-empty subsets, such that no edge joins two vertices in the same set.

Let X₁ and X₂ be two graphs. The join of X₁ and X₂, denoted by X₁+ X₂, is a graph such that V(X₁+ X₂) = V(X₁) U V(X₂) and E(X₁+ X₂) = E(X₁) U E(X₂) U {{x₁, x₂}|x₁ ∊ V(X₁),x₂ ∊ V(X₂)}. The lexicographic product of X₁ and X₂, denoted by X₁ [X₂], is a graph with vertex set V(X₁[X₂]) = V(X₁) × V(X₂), and edge set E(X₁[X₂]) = {{(x, y),(x₁, y₁)}|{x, x₁} ∈ E(X₁), or x = x₁and{y, y₁} ∈ E(X₂)}. The generalized lexicographic product of a graph G with a family of graphs {B_i|i ∈ V(G)}, denoted by G[B_i]_i∈V(G), is defined as a graph whose vertex set V(G[B_i]_i∈V(G)) = {(x, y_x)|x ∈ V(G), y_x ∈ V(B_x)}, and {(x, y),(x₁, y₁)} ∈ E (G[B_i]_i∈V(G)) if and only if {x, x₁} ∈ E(G), or x = x₁ and {y, y₁} ∈ E(B_x).

Let X and Y be graphs. A mapping f from V(X) to V(Y) is called a homomorphism (from X to Y) if {x₁, x₂} ∈ E(X) implies that {f(x₁), f(x₂)} ∈ E(Y). A homomorphism f from X to itself is called an endomorphism of X. Denote by End(X) the set of all endomorphisms of X. A retraction of a graph X is a homomorphism f from X to a subgraph Y of X such that the restriction f|_Y of f to V(Y) is the identity mapping on V(Y). It is known that the idempotents of End(X) are retractions of X Denote by Idpt(X) the set of all idempotents of End(X). Let f ∈ End(X). A subgraph of X is called the endomorphic image of X under f, denoted by I_f, if V(I_f) = f(V(X)) and {f(a), f(b)} ∈ E(I_f) if and only if there exist c ∈ f^–1(f(a)) and d ∈ f^–1(f(b)) such that {c, d} ∈ E(X). By ρ_f we denote the equivalence relation on V(X) induced by f, i.e., for a_; b ∈ V(X), (a, b) ∈ ρ_f if and only if f(a) = f(b). Denote by [a]_ρf the equivalence class containing a ∈ V(X) with respect to ρ_f.

An element a of a semigroup S is said to be regular if there exists x ∈ S such that axa = a. A semigroup S is called regular if all its elements are regular. A semigroup S is called orthodox if S is regular and the set of all idempotents forms a subsemigroup, that is, a regular semigroup is orthodox if the product of its any two idempotents is still an idempotent. A graph X is said to be End-regular(orthodox) if its endomorphism monoid End(X) is regular( orthodox). Clearly, End-orthodox graphs are End-regular.

For undefined notation and terminology in this paper the reader is referred to [1, 4, 5]. We list some known results which will be used in the sequel.

Lemma 1.1([16]). Let X be a graph and let f ∈ End(X). Then f is regular if and only if there exists g, h ∈ Idpt(X) such that ρ_g = ρ_f and I_h = I_f

Lemma 1.2([3]). If X [Y] is End-regular, then both X and Y are End-regular

Lemma 1.3([19]). If X is End-regular and Y is a retract of X, then Y is End-regular

Lemma 1.4([19]). Let X be a connected bipartite graph. Then X is End-regular if and only if X is one of the following graphs:

(1) Complete bipartite graph,

(2) Trees T with d(T) = 3,

(3) Cycle C₆and C8,

(4) Path with 5 vertices, i.e. P₅.

Lemma 1.5([9]). Let B₁and B₂be two connected bipartite graphs. Then B₁+ B₂is End-regular if and only if

(1) One of them is End-regular and the other is K₂, or

(2) B₁+ B₂= T₁+ T₂, where where T₁and T₂are trees with diameter 2, or

(3) B₁+ B₂ = K_m₁,_n₁ + K_m₂,_n₂, where K_{m_i},_{n_i} (i = 1_; 2) denotes complete bipartite graphs, or

(4) B₁+ B₂= T + K_{m, n}, where K_m;_n denotes a complete bipartite graph and T denotes a tree with diameter 2.

Lemma 1.6([10]). Let X and Y be two connected bipartite graphs. Then X[Y] is End-regular if and only if

(1) X = K₂and Y = K_{m, n}, where K_{m, n} denotes complete bipartite graphs, or

(2) X is End-regular and Y = K₂.

Lemma 1.7([3]). Let X be a connected bipartite graph. Then X is End-orthodox if and only if X is one of the following graphs : K₂, P₃, P₄, C₄.

Lemma 1.8([15]). Let G be a graph and let {B_i| i ∈ V(G)} be a family of graphs. If G[B_i]_i∈(G)is End-regular, then for any i ∈ V(G), Bi is End-regular

Lemma 1.9([10]). Let X and Y be two graphs. If X [Y] is End-orthodox, then both X and Y are End-orthodox

Lemma 1.10([9]). Let B₁and B₂be two connected bipartite graphs, then B₁+ B₂is End-orthodox if and only if

(1) One of them is End-orthodox and the other is K₂,

(2) B₁+ B₂= P₃C₄.

2 End-regular generalized lexicographic products of bipartite graphs

Recall that End-regular bipartite graphs are characterized in Lemma 1.4 and End-regular lexicographic products of bipartite graphs are determined in Lemma 1.6. In this section, we characterize the End-regular generalized lexicographic products of connected bipartite graphs. The following theorem is our main result.

Theorem2.1. Let G be a connected bipartite graph and let {B_i| i ∈ V(G)} be a family of connected bipartite graphs. Then G[B_i]_i∈V(G)is End-regular if and only if one of the following conditions hold:

(1) G = K₂with V(G) = {1,2}, B₁is End-regular and B₂= K₂;

(2) G = K₂with V(G) = {1, 2}, B₁and B₂are trees of diameter 2;

(3) G = K₂with V(G) = {1,2}, B₁and B₂are complete bipartite graphs;

(4) G = K₂with V(G) = {1_;2}, B₁is a tree of diameter 2 and B₂is a complete bipartite graph;

(6) G ≠ K₂is End-regular and B_i = K₂for any i ∈ V(G).

To prove our main result, we need the following characterizations of the regular endomorphisms of lexicographic products of bipartite graphs.

Lemma 2.2Let G be a bipartite graph and let {B_i|i ∈ V(G)} be a family of connected bipartite graphs. If G[B_i]_i∈V(G)is End-regular, then G is End-regular

Proof Since B_i is bipartite, its vertex set V(B_i) can be partitioned into two disjoint non-empty subsets A_i and C_i, such that no edge joins two vertices in the same set. Take y_i₀ ∈ A_i and z_i0 ∈ C_i such that {y_i₀, z_i₀} ∈ E(B_i) for any i ∈ V(G). Define the mapping on the vertex set V(B_i) as follows:

gi(x)={yi0,x∈Ai,zi0,x∈Ci.

Let f ∈ End(G). Define a mapping F from V(G[B_i]_i∈V(G)) to itself by

F((i,x))={(f(i),yf(i)0),x∈Ai,(f(i),zf(i)0),x∈Ci.

Let (i, x_i), (j, x₂) ∈ V(G[B_i]_i∈V(G)) with {(i, x_i), (j, x₂)} e E(G[Bi]_isViG)). If {i, j} ∈ E(G), then {f(i), f(j)} ∈ E(G) and so {F((i, x₁)), F((j, x₂))} ∈ E(G[B_i]_i∈V(G)). If i = j and{x₁, x₂} ∈ E(B_i), then {F((i, x₁)), F((j, x₂))} = {(f (i), y_f_(i)0),(f(i), z_f_(i)0)} ∈ E(G[B_i]_i∈V(G)). Hence F ∈ End(G[B_i]_i∈V(G)). Note that F is regular. Thus there exists pseudoinverse K ∈ End(G[B_i]_i∈V(G)) such that FKF D F.

Let (i, x) ∈ V(G[B_i]_i∈V(G)) and K((i, x)) = (j, x′) for some j ∈ V(G) and x′ ∈ V(B_j). If x ∈ A_i, then x′ ∈ A_j; If x ∈ C_i, then x′ ∈ C_j Define a mapping k from V(G) to itself by k(i) = j, where j satisfies K((i, x)) = (j, x′) for some x′ ∈ V(B_j). We now prove that k ∈ End(G). Let i, j ∈ V(G) be such that {i, j} ∈ E(G). Then {(i, x₁), (j, x₂)} ∈ E(G[B_i]_i∈V(G)) for any x₁ ∈ A_i and x₂ ∈ A_j Since K ∈ End{G[B_i]_i∈EV(G)), there exists x1' ∈ V(B_k_(i)) and x2' ∈ V(B_k_(j)) such that {K((i, x₁)), K((j, x₂))} = {{k{i), x1'), (k(j), x2')} ∈ E(G[B_i]_i∈V(G)). If {k(i), k(j)} ∉ E(G), then k(i) = k(j) and {x1', x2'} ∈ E(B_k_(i)). Note that x1', x2' ∈ A_k_(i). This is a contradiction. Hence {k(i), k(j)} ∉E(G) and so k ∈ End(G). And by the definition of k, we get that there exists x′ ∈ V(B_fkf_(i)) such that FKF((i, x)) = (fkf(i), x′). Hence fkf = f This means that k is a pseudoinverse of f and so f is regular. Therefore G is End-regular. □

Lemma 2.3Let G be a bipartite graph and let {B_i|i ∈ V(G)} be a family of connected bipartite graphs. If G₁is a retract of G and {H_i|i ∈ V(G)} are retracts of {B_i|i ∈ V(G)}, thenG₁[H_i]_i∈V(G₁)is a retract of G[B_i]_i∈V(G).

Proof. We only need to show that there exists a retraction from G[B_i]_i∈V(G) to G₁[H_i]_i∈V(G₁). Since G₁ and {H_i|i ∈ V(G)} are retracts of G and {Bi| i ∈ V(G)} respectively, there exist retractions f from G to G₁ and h_i from B_i to H_i For any j ∈ V(G₁), take y_j₁ ∈ A_j ⋂ I_{h_j} and Zj₁ ∈ C_j ⋂ I_{h_j}.

Let F be the mapping from G[B_i]_i∈V(G) to itself defined as follows:

F((k,y))={(k,hk(y)),if k∈V(G1),(f(k),yf(k)1),if k∉V(G1)and y∈Ak,(f(k),zf(k)1),if k∉V(G1)and y∈Ck.

Let (j, x₁), (k, x₂) ∈ V(G[B_i]_i∈V(G)) be such that {(j, x₁), (k, x₂)} ∈ E(G[B_i]_i∈V(G)). Then {j, k} ∈ or j = k and {x₁, x₂} ∈ E(B_j). If {j, k} ∈ E(i), then there exist x1' ∈ V(B_f_(j)) and x2' ∈ V(B_f_(k)) such that {F((j, x₁)), F((k, x₂))} = {(f(j), x1', (f(k), x2'} ∈ E(G[B_i]_i∈V(G)). If j = k and {x₁,x₂} ∈ E(B_j), without loss of generality, suppose x₁ ∈ A_j and x₂ ∈ C_j. If j ∈ V(G₁), then {F((j, x₁)), F((k, x₂))} = {(j, h_j (x₁)), (j, h_j (x₂))}. Since h_j is a retraction of B_j, {h_j (x₁), h_j (x₂)} ∈ E(B_j). Hence {(j, h_j (x₁)), (j, h_j (x₂))} ∈ E(G[B_i]_i∈V(G)). If j ∉ V(G₁), then {F((j, x₁)), F((k, x₂))} = {(f(j), y_f_(j)1), (f(j), z_f_(j)1)} ∈ E(G[B_i] _i∈V(G)). Therefore F ∈ End(G[B_i]_i∈V(G)). It is easy to check that I_F = G₁[H_i]_i∈V(G₁). Let (j, x) ∈ V(G₁[H_i]_i∈V(G₁). Then F((j, x)) = (j, h_j(x)) = (j, x) and so F is a retraction from G[B_i]_i∈V(G) to G₁[H_i]_i∈V(G₁). □

Next we start to seek the conditions for a generalized lexicographic product of bipartite graphs under which G[B_i]_i∈V(G) is End-regular.

Lemma 2.4Let G = K₂with V(G) = {1,2} and let {B_i|i ∈ V(G)} be two connected bipartite graphs. Then G[B_i]_i∈V(G)is End-regular if and only if one of the following conditions hold:

(1) B₁is End-regular and B₂= K₂,

(2) Both B₁and B₂are trees of diameter 2,

(3) Both B₁and B₂are complete bipartite graphs,

(4) B₁is a tree of diameter 2 and B₂is a complete bipartite graph

Proof Note that G[B_i]_i∈V(G) is isomorphic to B₁+ B₂. Then the result follows directly from Lemma 1.5. □

Lemma 2.5Both P₃ [P₃, K₂, K₂] and P₃ [K₂, P₃, K₂] are not End-regular

$Fig. 1 Graph P3[P3, K2, K2]$

Fig. 1

Graph P₃[P₃, K₂, K₂]

$Fig. 2 Graph P3[K2, P3, K2]$

Fig. 2

Graph P₃[K₂, P₃, K₂]

Proof Let

f=(x1x2x3y1y2z1z2x1y1x1y2x2x3y1).

Then f ∈ End(P₃[P₃, K₂, K₂]). It is easy to see that ρ_f = {[x₁, x₃], [y₂], [z₁], [x₂, z₂], [y₁]}. Suppose that there exists an idempotent endomorphism g of P₃[P₃, K₂, K₂] such that ρ_g = ρ_f Then g(y₁) = y₁, g(y₂) = y₂ and g(z₁) = z₁. Since z₂ is adjacent to every vertices of {y₁, y₂, z₁}, g(z₂) is adjacent to every vertex of g({y₁, y₂, z₁}) = {y₁, y₂, z₁}. Note that z₂ is the only vertex in V(P₃[P₃, K₂, K₂]) adjacent to every vertices of {y₁, y₂, z₁}. Then g(x₂) = g(z₂) = z₂. Since {x₁, x₂} ∈ E(P₃[P₃, K₂, K₂]), {g(x₁) g(x₂)} = {g(x₁), z₂)} ∈ E(P₃[P₃, K₂, K₂]). Thus g(x₁) ∈ {y₁, y₂, z₁)}. This is a contradiction. By Lemma 1.1 f is not regular and so P₃[P₃, K₂, K₂] is not End-regular.

Let

h=(a1a2b1b2b3c1c2a1b1a2b2a2b3a1).

Then h ∈ End(P₃[K₂, P₃, K₂]). It is easy to see that ρ_h = {[a₁, c₂], [a₂], [b₁, b₃], [b₂], [c₁]}. Suppose that there exists an idempotent endomorphism k of P₃[K₂, P₃, K₂] such that ρ_h = ρ_h Then k(a₂) = a₂, k(b₂) = b₂ and k(c₁) = c₁. Since b₁ is adjacent to every vertex of {a₂, b₂, c₁}, k(b₁) is adjacent to every vertex of k({a₂, b₂,c₁}) = {a₂, b₂, c₁}. Note that {a₁, a₂} ∈ E(P₃[K₂, P₃, K₂]), {c₂, b₂} ∈ E(P₃[K₂, P₃, K₂]) and {c₂, c₁} ∈ E(P₃[K₂, P₃, K₂]). Then k(a₁) = k(c₂) is adjacent to every vertex of k({a₂, b₂, c₁}) = {a₂, b₂, c₁}. It follows from {a₁, b₁} ∈ E(P₃[K₂, P₃, K₂]) that {f(a₁), f(b₁)} ∈ E(P₃[K₂, P₃, K₂]). Note that there are no two adjacent vertices in V(P₃[K₂, P₃, K₂]) adjacent to every vertex of {a₂, b₂, c₁}. This is a contradiction. By Lemma 1.1 h is not regular and so P₃[K₂, P₃, K₂] is not End-regular. □

Lemma 2.6Let G ≠ K₂be a connected End-regular bipartite graph and let {B_i|i ∈ V(G)} be a family of connected bipartite graphs. Then G[B_i]_i∈V(G)is End-regular if and only if B_i = K₂for any i ∈ V(G).

Proof. Necessity. Suppose that G[B_i]_i∈V(G) is End-regular. Then G is End-regular by Lemma 2.2 and B_i is End-regular for any i ∈ V(G) by Lemma 1.8. Note that G is a connected bipartite graph and G ≠ K₂. Then G contains at least two edges. Thus P₃ is a retract of G Let {B_i|i ∈ V(G)} be a family of connected bipartite graphs. If there exists j ∈ V(G) such that B_j ≠ K₂, then P₃ is a retract of B_j Note that K₂ is a retract of any bipartite graphs. By Lemma 2.3, P₃[P₃, K₂, K₂] or P₃[K₂, P₃, K₂] is a retract of G[Bi]_i∈V(G). Since both P₃[P₃, K₂, K₂] and P₃[K₂, P₃, K₂] are not End-regular, by Lemma 1.3, G[B_i]_i∈V(G) is not End-regular. This is a contradiction.

Sufficiency. If B_i = K₂ for any i ∈ V(G), then G[B_i]_i∈V(G)= G[K₂], the lexicographic products of G and K₂. By Lemma 1.6, G[B_i]_i∈V(G) is End-regular. □

With these preparations, the proof of our main result is straightforward.

Proof of Theorem 2.1 This follows directly from Lemmas 2.4 and 2.6. □

3 End-orthodox generalized lexicographic products of bipartite graphs

In this section, we characterize the End-orthodox generalized lexicographic products of bipartite graphs. Since End-orthodox graphs are End-regular, we always assume our graphs are End-regular in this section. The following theorem is our main result.

Theorem 3.1Let G be a connected bipartite graph and let {B_i|i ∈ V(G)} be a family of connected bipartite graphs. Then G[B_i]_i∈V(G)is End-orthodox if and only if

(1) G = K₂, B₁= K₂and B₂ ∈ {K₂, P₃, P₄, C₄}, or

(2) G = K₂, B₁= P₃and B₂=C₄.

To prove our main result, we need the following characterizations of the endomorphism monoids of lexicographic products of bipartite graphs.

Lemma 3.2If X is End-orthodox and Y is a retract of X, then Y is End-orthodox

Proof. Since X is End-orthodox, X is End-regular. Let Y be a retract of X By Lemma 1.3, Y is End-regular. To show that Y is End-orthodox, we only need to prove that the composition of any two idempotent endomorphisms of Y is also idempotent.

Since Y is a retract of X, there exists a retraction f from X to Y Let g₁,g₂ ∈ Idpt(Y). Then g₁f, g₂f ∈ Idpt(X). Moreover, (g₁f)(y) = g₁(y) and (g₂f)(y) = g₂(y) for any y ∈ V(Y). Now it is easy to see that [(g₁f)(g₂f)](y) = (g₁f)[(g₂f)(y)] = (g₁f)(g₂(y)). Note that g₂(y) ∈ V(Y). Thus (g₁f)(g₂(y)) = g₁(g₂(y)) = (g₁g₂)(y) ∈ V(Y). Hence [(g₁f)(g₂f)]²(y) = [(g₁f)(g₂f)]([(g₁f)(g₂f)](y)) = [(g₁f)(g₂f)][(g₁g₂)(y)] = (g₁g₂)[(g₁g₂)(y)] = (g₁g₂)²(y). Note that X is End-orthodox. Then [(g₁f)(g₂f)]²= (g₁f)(g₂f). Thus (g₁g₂)²(y) = (g₁g₂)(y) for any y ∈ V(Y). Hence g₁g₂ ∈Idpt(Y) and so Y is End-orthodox. □

Lemma 3.3Let G be a bipartite graph and let {B_i|i ∈ V(G)} be a family of bipartite graphs. If G[B_i]_i∈V(G)is End-orthodox, then G and B_i are End-orthodox for any i ∈ V(G).

Proof Since G[B_i]_i∈V(G) is End-orthodox, G[B_i]_i∈V(G) is End-regular. Then G is End-regular by Lemma 2.2 and B_i are End-regular for any i ∈ V(G) by Lemma 1.8.

Since B_i is bipartite, its vertex set V(B_i) can be partitioned into two disjoint non-empty subsets A_i and C_i, such that no edge joins two vertices in the same set. Take y_i₀ ∈ A_i and z_i₀ ∈ C_i for any i ∈ V(G). Let F be a mapping from V(G[B_i]_i∈V(G)) to itself defined by

F((i,x))={(i,yi0),x∈Ai,(i,zi0),x∈Ci.

Then it is easy to check that F ∈ Idpt(G[B_i]_i∈V(G)) and I_F ≅ G[K₂]. Note that G[K₂] is a retract of G[B_i]_i∈V(G). By Lemma 3.3, G[K₂] is End-orthodox. By Lemma 1.9, G is End-orthodox.

Let i ∈ V(G) and g₁, g₂ ∈ Idpt(B_i). Define two mappings G₁ and G₂ from V(G[B_i]_i∈V(G)) to itself by

G1((i,x))={(i,g1(x)),x∈V(Bi),(i,x),others. and G2((i,x))={(i,g2(x)),x∈V(Bi),(i,x),others.

Then it is easy to check that G₁, G₂ ∈ Idpt(G[B_i]_i∈V(G)) and so G₁G₂ is also an idempotent of End(G[B_i]_i∈V(G)) since G[B_i]_i∈V(G) is End-orthodox. For any x ∈ V(B_j), we have

(G1G2)2((i,x))=(G1G2)((i,(g1g2)(x)))=(i,(g1g2)2(x))=(G1G2)((i,x))=(i,(g1,g2)(x)).

Clearly, (g₁g₂)²= g₁g₂. Hence g₁g₂ ∈ Idpt(B_i) and so B_i is End-orthodox. □

Proof of Theorem 3.1 Sufficiency. In case (1), G[B_i]_i∈V(G) is isomorphic to K₂+ K₂, K₂+ P₃, K₂+ P₄, or K₂+ C₄. By Lemma 1.10, we have immediately that G[B_i]_i∈V(G) is End-orthodox. In case (2), G[B_i]_i∈V(G) is isomorphic to P₃+ C₄. Also by Lemma 1.10, G[B_i]_i∈V(G) is End-orthodox.

Necessity. We only need to show that G[B_i]_i∈V(G) is not End-orthodox for the following cases:

Case 1. G = K₂, B₁ and B₂ do not satisfy the conditions (1) and (2). Then G[B_i]_i∈V(G)= B₁+ B₂. By Lemma 1.10, G[B_i]_i∈V(G) is not End-orthodox.

$Fig. 3 Graphs P3[K2], P4[K2] and C4[K2]$

Fig. 3

Graphs P₃[K₂], P₄[K₂] and C₄[K₂]

Case 2. G = P₃ and B_i = K₂ for any i ∈ V(G). Then G[B_i]_i∈V(G) ≅ P₃ [K₂] (see Fig. 3). We will show that there exist f, g ∈ Idpt(G[B_i]_i∈V(G)) such that fg ∉ Idpt(G[B_i]_i∈V(G)). Let

f=(x1x2y1y2z1z2z2z1y1y2z1z2)

and

g=(x1x2y1y2z1z2x1x2y1y2x1x2).

Then f, g ∈ Idpt(G[B_i]_i∈V(G)). But

fg=(x1x2y1y2z1z2z2z1y1y1z2z1).

It is easy to check fg ∉ Idpt(G[B_i]_i∈V(G)). Hence G[B_i]_i∈V(G) is not End-orthodox.

Case 3.G = P₄ and B_i = K₂ for any i ∈ V(G). Then G[B_i]_i∈V(G) ≅ P₄[K₂] (see Fig.3). Let

f=(a1a2b1b2c1c2d1d2c1c2d1d2c1c2d1d2)

and

g=(a1a2b1b2c1c2d1d2a1a2b1b2a2a1b1b2).

Then f, g ∈ Idpt(G[B_i]_i∈V(G)). But

fg=(a1a2b1b2c1c2d1d2c1c2d1d2c2c1d1d2).

It is easy to check fg ∉ Idpt(G[B_i]_i∈V(G)). Hence G[B_i]_i∈V(G) is not End-orthodox.

Case 4. G = C₄ and B_i = K₂ for any i ∈ V(G). Then G[B_i]_i∈V(G) ≅ C₄[K₂] (see Fig.3). Let

f=(a1a2b1b2c1c2d1d2a1a2b1b2a1a2b1b2)

and

g=(a1a2b1b2c1c2d1d2a1a2d2d1a1a2d1d2).

Then f, g ∈ Idpt(G[B_i]_i∈V(G)). Bu

fg=(a1a2b1b2c1c2d1d2a1a2b2b1a1a2b1b2).

It is easy to check fg ∉ Idpt(G[B_i]_i∈V(G)). Hence G[B_i]_i∈V(G) is not End-orthodox. □

Acknowledgement

The authors wish to express their gratitude to the referees for their helpful suggestions and comments.

This research was partially supported by the National Natural Science Foundation of China(No. 11301151), the Subsidy Scheme of Young Teachers of Henan Province(No.2014GGJS-057) and the Innovation Team Funding of Henan University of Science and Technology(NO.2015XTD010).

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Received: 2015-9-5

Accepted: 2016-2-3

Published Online: 2016-4-6

Published in Print: 2016-1-1

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