The Restricted Edge-Connectivity of Strong Product Graphs

The restricted edge-connectivity of a connected graph $G$, denoted by $\lambda^{\prime}(G)$, if it exists, is the minimum cardinality of a set of edges whose deletion makes $G$ disconnected and each component with at least 2 vertices. It was proved that if $G$ is not a star and $|V(G)|\geq4$, then $\lambda^{\prime}(G)$ exists and $\lambda^{\prime}(G)\leq\xi(G)$, where $\xi(G)$ is the minimum edge-degree of $G$. Thus a graph $G$ is called maximally restricted edge-connected if $\lambda^{\prime}(G)=\xi(G)$; and a graph $G$ is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of $G$. The strong product of graphs $G$ and $H$, denoted by $G\boxtimes H$, is the graph with vertex set $V(G)\times V(H)$ and edge set $\{(x_1,y_1)(x_2,y_2)\ |\ x_1=x_2$ and $y_1y_2\in E(H)$; or $y_1=y_2$ and $x_1x_2\in E(G)$; or $x_1x_2\in E(G)$ and $y_1y_2\in E(H)$\}. In this paper, we determine, for any nontrivial connected graph $G$, the restricted edge-connectivity of $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$, where $P_n$, $C_n$ and $K_n$ are the path, the cycle and the complete graph on $n$ vertices, respectively. As corollaries, we give sufficient conditions for these strong product graphs $G\boxtimes P_n$, $G\boxtimes C_n$ and $G\boxtimes K_n$ to be maximally restricted edge-connected and super restricted edge-connected.


Introduction
For notations and graph-theoretical terminology not defined here, we follow [2].All graphs in this paper are undirected, simple and finite.Let G = (V, E) be a graph, where V = V (G) is the vertex set and E = E(G) is the edge set.The order of G is |V (G)|, and the size of For two nonempty subsets X, Y ⊆ V (G), [X, Y ] G denotes the set of edges with one end in X and the other in Y .When Y = V (G)\X, the set [X, Y ] G is called an edge-cut of G associated with X.The edge-connectivity λ(G) of a graph G is defined as the cardinality of a minimum edge-cut of G.It is well known that λ(G) ≤ δ(G).Thus a graph G is said to be maximally edge-connected if λ(G) = δ(G); and a graph G is said to be super edge-connected if each minimum edge-cut isolates a vertex of G.A super edge-connected graph must be maximally edge-connected.But the converse is not true.For example, the cycle C n (n ≥ 4) is maximally edge-connected but not super edge-connected.
As an interconnection network can be modeled by a graph, the edge-connectivity can be used to measure the network reliability.But there is a deficiency, which allows all edges incident with a vertex to fail simultaneously.This situation is highly improbable in practical network applications.For compensating this deficiency, Esfahanian and Hakimi [6] introduced the notion of restricted edge-connectivity.If an edge set S ⊆ E(G) satisfies G − S is disconnected and each component of G − S has at least 2 vertices, then S is called a restricted edge-cut.If G has at least one restricted edge-cut, then the restricted edge-connectivity of G, denoted by λ ′ (G), is the cardinality of a minimum restricted edge-cut of G.It was proved in [6] that if G is not a star and its order is at least four, then λ ′ (G) ≤ ξ(G).Thus, if λ ′ (G) = ξ(G), then G is said to be maximally restricted edge-connected; if each minimum restricted edge-cut isolates an edge of G, then G is said to be super restricted edge-connected.A super restricted edge-connected graph must be maximally restricted edge-connected.But the converse is not true.For example, the cycle C n (n ≥ 6) is maximally restricted edge-connected but not super restricted edge-connected.
The concept of graph product is utilized to construct larger graphs from smaller ones.There exist various kinds of graph products, including Cartesian product, direct product and strong product, etc.Given two graphs G and H, the vertex sets of the Cartesian product G✷H, the direct product G × H and the strong product G ⊠ H are all V (G) × V (H).For two distinct vertices (x 1 , y 1 ) and (x 2 , y 2 ), they are adjacent in G✷H if and only if x 1 = x 2 and y 1 y 2 ∈ E(H), or y 1 = y 2 and x 1 x 2 ∈ E(G); they are adjacent in G × H if and only if x 1 x 2 ∈ E(G) and y 1 y 2 ∈ E(H); and they are adjacent in G ⊠ H if and only if x 1 = x 2 and y 1 y 2 ∈ E(H), or y 1 = y 2 and x 1 x 2 ∈ E(G), or x 1 x 2 ∈ E(G) and y 1 y 2 ∈ E(H).Clearly, E(G⊠H) = E(G✷H)∪E(G×H).
In [7], Klavžar and Špacapan determined the edge-connectivity of the Cartesian product of two nontrivial graphs.Shieh [13] characterized the super edge-connected Cartesian product graphs of two maximally edge-connected regular graphs.For the results on the restricted edgeconnectivity of Cartesian product graphs, see [8,9,11] for references.Some bounds on the edge-connectivity of the direct product of graphs were given by Brešar and Špacapan [4].The edge-connectivity of the direct product of a nontrivial graph and a complete graph was obtained by Cao, Brglez, Špacapan and Vumar [5].In [14], Špacapan not only determined the edge-connectivity of the direct product of two general graphs, but also characterized the structure of each minimum edge-cut in these direct product graphs.In [10], Ma, Wang and Zhang studied the restricted edge-connectivity of the direct product of a nontrivial graph with a complete graph.In [1], Bai, Tian and Yin further studied the super restricted edge-connectedness of these direct product graphs.
In [3], Brešar and Špacapan determined the edge-connectivity of the strong products of two connected graphs.Ou and Zhao [12] studied the restricted edge-connectivity of strong product of two triangle-free graphs.In [15], Wang, Mao, Ye and Zhao gave an expression of the restricted edge-connectivity of the strong product graphs with two maximally restricted edge-connected graphs.
Motivated by the results above, we will study the restricted edge-connectivity of the strong product of a nontrivial connected graph with a path, or a cycle, or a complete graph in this paper.As corollaries, we give sufficient conditions for these strong product graphs to be maximally restricted edge-connected and super restricted edge-connected.In the next section, we will introduce some definitions and lemmas.The main results will be presented in Section 3.

Preliminary
Denote by P n , C n and K n the path, the cycle and the complete graph on n vertices, respectively.
Let G and H be two graphs.Define a natural projection p on V (G) × V (H) as follows: p(x, y) = y for any (x, y) ∈ V (G) × V (H).For any given x ∈ V (G), the subgraph induced by {(x, y)|y ∈ V (H)} in G⊠H, denoted by H x .Analogously, for any given y ∈ V (H), the subgraph induced by The edge-connectivity of the strong product of two connected nontrivial graphs was given in the following lemma.
with all equalities holding if and only if X is a minimum-degree vertex.
Lemma 2.5.( [12]) Let G and H be two connected graphs.Then

Main results
Theorem 3.1.Let G be a connected nontrivial graph with m vertices.Then Thus, by (1), we have By Lemma 2.3 and (1), we have For any (x, If δ(G) ≥ 3, then by ( 2) and (3), we have . Thus, by (1), we have ] is connected, then, by a similar argument as Subcase 2.1, we can also obtain This proof is thus complete.
Analogously, if each vertex Case 2. There exists a vertex x a ∈ V (G) and a vertex )) = {y s+1 , y s+2 , . . ., y s+k }, where the addition is modular n operation.Without loss of generality, assume then by lemma 2.3 and (4), we have By Lemma 2.3 and (4), we have 3) and ( 5), we have This proof is thus complete.
Analogously, if each vertex Case 2. There exists a vertex x a ∈ V (G) and a vertex By the assumption, we know V (K xa n ) and V (G y b ) are contained in D 2 .Since any two distinct vertices are adjacent in K n , by renaming the vertices of V (K n ), we can let p(V (D 1 )) = {y s+1 , y s+2 , . . ., y s+k }.Furthermore, assume s + k < b.For 1 . By the definition of the strong product, for any y ∈ Y , we have

By lemma 2.2, we have
Subcase 2.2.2.There are at least two integers in {1, . . ., k}, say 1 and 2, such that For an edgee = uv ∈ E(G), ξ G (e) = d G (u) + d G (v) − 2 is the edgedegree of e in G.The minimum edge-degree of G, denoted by ξ(G), is min{d G (u)+d G (v)−2 | e = uv ∈ E(G)}.Obviously, ξ(G) ≥ 2δ(G) − 2,with the equality holding if and only if there is an edgee = uv ∈ E(G) such that d G (u) = d G (v) = δ(G).For a vertex set A ⊆ V (G), the induced subgraph of A in G, denoted by G[A],is the graph with vertex set A and two vertices u and v in A are adjacent if and only if they are adjacent in G.

Case 2 .
There exists a vertex x a ∈ V (G) and a vertex y b ∈ V (P n ) such that P xa n ∩ D 1 = ∅ and G y b ∩ D 1 = ∅, or P xa n ∩ D 2 = ∅ and G y b ∩ D 2 = ∅.Without loss of generality, assume P xa n ∩ D 1 = ∅ and G y b ∩ D 1 = ∅.By the assumption, we know V (P xa n ) and V (G y b ) are contained in D 2 .Let p(V (D 1 )) = {y s+1 , y s+2 , . . ., y s+k }.Without loss of generality, assume s + k < b.For 1
is the strong product of the complete graph with only one vertex a and H, and {b} ⊠ H is the strong product of the complete graph with only one vertex b and H.It is not difficult to see that K 2 ⊙ H is connected if and only if H is connected.Let H be a connected graph and S be an edge cut of K 2 ⊙H, where V (K 2 ) = {a, b}.If the vertices of {a} ⊠ H are in different components of K 2 ⊙ H − S as well as {b} ⊠ H, then |S| ≥ 2λ(H).