The $g$-extra connectivity of the strong product of paths and cycles

Let $G$ be a connected graph and $g$ be a non-negative integer. The $g$-extra connectivity of $G$ is the minimum cardinality of a set of vertices in $G$, if it exists, whose removal disconnects $G$ and leaves every component with more than $g$ vertices. The strong product $G_1 \boxtimes G_2$ of graphs $G_1=(V_{1}, E_{1})$ and $G_2=(V_{2}, E_{2})$ is the graph with vertex set $V(G_1 \boxtimes G_2)=V_{1} \times V_{2}$, where two distinct vertices $(x_{1}, x_{2}), (y_{1}, y_{2}) \in V_{1} \times V_{2}$ are adjacent in $G_1 \boxtimes G_2$ if and only if $x_{i}=y_{i}$ or $x_{i} y_{i} \in E_{i}$ for $i=1, 2$. In this paper, we obtain the $g$-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.


Introduction
Let G be a graph with vertex set V (G) and edge set E(G).The minimum degree of G is denoted by δ(G).A vertex cut in G is a set of vertices whose deletion makes G disconnected.The connectivity κ(G) of the graph G is the minimum order of a vertex cut in G if G is not a complete graph; otherwise κ(G) = |V (G)| − 1. Usually, the topology structure of an interconnection network can be modeled by a graph G, where V (G) represents the set of nodes and E(G) represents the set of links connecting nodes in the network.Connectivity is used to measure the reliability the network, while it always underestimates the resilience of large networks.
To overcome this deficiency, Harary [9] proposed the concept of conditional connectivity.For a graph-theoretic property P, the conditional connectivity κ(G; P) is the minimum cardinality of a set of vertices whose deletion disconnects G and every remaining component has property P. Later, Fàbrega and Fiol [7] introduced the concept of g-extra connectivity, which is a kind of conditional connectivity.Let g be a non-negative integer.A subset S ⊆ V (G) is called a g-extra cut if G− S is disconnected and each component of G− S has at least g + 1 vertices.The g-extra connectivity of G, denoted by κ g (G), is the minimum order of a g-extra cut if G has at least one g-extra cut; otherwise define κ g (G) = ∞.If S is a g-extra cut in G with order κ g (G), then we call S a κ g -cut.Since κ 0 (G) = κ(G) for any connected graph G that is not a complete graph, the g-extra connectivity can be seen as a generalization of the traditional connectivity.The authors in [4] pointed out that there is no polynomial-time algorithm for computing κ g for a general graph.Consequently, much of the work has been focused on the computing of the g-extra connectivity of some given graphs, see [1,4,6,8,[10][11][16][17][18][19][20][21] for examples.
The most studied four standard graph products are the Cartesian product, the direct product, the strong product and the lexicographic product.The Cartesian product of two graphs G 1 and G 2 , denoted by G 1 G 2 , is defined on the vertex sets V (G 1 ) × V (G 2 ), and (x 1 , y 1 )(x 2 , y 2 ) is an edge in G 1 G 2 if and only if one of the following is true: (i) if and only if one of the following holds: (i) Špacapan [13] proved that for any nontrivial graphs Lü, Wu, Chen and Lv [12] provided bounds for the 1-extra connectivity of the Cartesian product of two connected graphs.Tian and Meng [15] determined the exact values of the 1-extra connectivity of the Cartesian product for some class of graphs.In [5], Chen, Meng, Tian and Liu further studied the 2-extra connectivity and the 3-extra connectivity of the Cartesian product of graphs.
Brešar and Špacapan [3] determined the edge-connectivity of the strong products of two connected graphs.For the connectivity of the strong product graphs, Špacapan [14] obtained T heorem 1.1 in the following.Let S i be a vertex cut in G i for i = 1, 2, and let A i be a component of G i − S i for i = 1, 2. Following the definitions in [14], Theorem 1.1.( [14]) Let G 1 and G 2 be two connected graphs.Then every minimum vertex cut in G 1 ⊠ G 2 is either an I-set or an L-set in G 1 ⊠ G 2 .
Motivated by the results above, we will study the g-extra connectivity of the strong product graphs.In the next section, we introduce some definitions and lemmas.In Section 3, we will give the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.Conclusion will be given in Section 4.

Preliminary
For graph-theoretical terminology and notations not defined here, we follow [2].Let G be a graph with vertex set V (G) and edge set E(G).The neighborhood of a vertex . We use P n to denote the path with order n and C n to denote the cycle with order n.
Let G 1 and G 2 be two graphs.Define two natural projections p 1 and p 2 on V (G 1 )×V (G 2 ) as follows: p 1 (x, y) = x and p 2 (x, y) = y for any (x, y) ) by x S, and analogously, for any y ∈ V (G 2 ), denote S ∩ V (G 1y ) by S y .Furthermore, we use x S = V ( x G 2 ) \ x S and S y = V (G 1y ) \ S y .By a similar argument as the proof of the second paragraph of T heorem 3.2 in [14], we can obtain the following lemma.
Lemma 2.1.Let G be the strong product G 1 ⊠ G 2 of two connected graphs G 1 and G 2 , and let g be a non-negative integer.Assume G has g-extra cuts and S is a κ g -cut of G.
, we find that (x, y 1 ) ∈ x S, moreover, for any (x, u) ∈ x S, we find that (x, u) is not adjacent to (x, y 1 ), otherwise, (x, u) would be adjacent to (x 1 , y 1 ), which is not true since those two vertices are in different components of

Main results
Let H be a subgraph of G 1 ⊠ G 2 .For the sake of simplicity, we use we assume m, n ≥ 3 in the following theorem.Theorem 3.1.Let g be a non-negative integer and G. We consider two cases in the following.
Case 1. x S = ∅ for all x ∈ V (P m ), or S y = ∅ for all y ∈ V (P n ).
Case 2. There exist a vertex x a ∈ V (P m ) and a vertex y b ∈ V (P n ) such that xa S = S y b = ∅.

By the assumption
1 and the theorem holds.Thus, we only need to show that in the remaining proof.Let (x s+i , y d i ) be the vertex in x s+i H such that d i is maximum for i = 1, • • • , k, and let (x r j , y t+j ) be the vertex in H y t+j such that r j is maximum for j For the convenience of counting, we will construct an injective mapping Although D and R may have common elements, we consider the elements in D and R to be different in defining the mapping f below.
First, the mapping f on D is defined as follows.
Let (x s+i , y d i ) be the vertex in x s+i H such that d i is maximum for i = 1, • • • , k, and let (x l j , y t+j ) and (x r j , y t+j ) be the vertices in H y t+j such that l j and r j are listed in the foremost and in the last along the sequence (a For the convenience of counting, we will construct an injective ), (x s+k+1 , y d k +1 )}.Although D, L and R may have common elements, we consider the elements in D, L and R to be different in defining the mapping f below.
First, the mapping f on D is defined as follows.
If (x r j , y t+j ) satisfies (x r j +1 , y t+j ) / ∈ F 1 for any j ∈ {1, • • • , h}, then we are done.Otherwise, for each (x r j ′ , y t+j ′ ) satisfying (x r j ′ +1 , y t+j ′ ) ∈ F 1 , we give the definition as follows.By the definitions of D and R, we have ), respectively.The mapping f on R is defined well .
Finally, we construct an injective mapping The proof is thus complete.
Since κ g (C 3 ⊠ C n ) = 6 for g ≤ 3⌊ n−2 2 ⌋ − 1, we assume m, n ≥ 4 in the following theorem.Theorem 3.3.Let g be a non-negative integer and Proof.Denote C m = x 0 x 1 . . .x m−1 x m (where x 0 = x m ) and C n = y 0 y 1 . . .y n (where y 0 = y n ).The addition of the subscripts of x in the proof is modular m arithmetic, and the addition of the subscripts of y in the proof is modular n arithmetic.Let G. We consider two cases in the following.
definitions of T and R, we have {(x r j ′ , y t+j ′ ), • • • , (x r j ′ , y d t j ′ −s )} ⊆ R. Now, we define f ((x r j ′ , y t+j ′ )) = (x r j ′ +1 , y t+j ′ −1 ) and change the images of (x r j ′ , y t+j Note that the proof of four paragraphs above gives the definition of the mapping f on R. In the following proof, we will give the definition of the mapping f on L. Fifth, for each vertex (x l j , y t+j ) satisfying (x l j −1 , y t+j ) / ∈ F 1 , define f ((x l j , y t+j )) = (x l j −1 , y t+j ).
If (x l j , y t+j ) satisfies (x l j −1 , y t+j ) / ∈ F 1 for any j ∈ {1, • • • , h}, then we are done.Otherwise, for each (x l j ′ , y t+j ′ ) satisfying (x l j ′ −1 , y t+j ′ ) ∈ F 1 , we define as follows.By the definitions of D and L, we have Sixth, for each vertex (x l j , y t+j ) satisfying (x l j −1 , y t+j ) / ∈ F 2 , define f ((x l j , y t+j )) = (x l j −1 , y t+j ).

Conclusion
Graph products are used to construct large graphs from small ones.Strong product is one of the most studied four graph products.As a generalization of traditional connectivity, g-extra connectivity can be seen as a refined parameter to measure the reliability of interconnection networks.There is no polynomial-time algorithm to compute the g (≥ 1)-extra connectivity for a general graph.In this paper, we determined the g-extra connectivity of the strong product of two paths, the strong product of a path and a cycle, and the strong product of two cycles.In the future work, we would like to investigate the g-extra connectivity of the strong product of two general graphs.

Figure 1 .
Figure 1.An illustration for the proof of Theorem 3.1.