Extra Connectivity of Strong Product of Graphs

The $g$-$extra$ $connectivity$ ${\kappa }_g(G)$ of a connected graph $G$ is the minimum cardinality of a set of vertices, if it exists, whose deletion makes $G$ disconnected and leaves each remaining component with more than $g$ vertices, where $g$ is a non-negative integer. The $strong$ $product$ $G_1⊠G_2$ of graphs $G_1$ and $G_2$ is the graph with vertex set $V(G_1⊠G_2)=V(G_1)\times V(G_2)$, where two distinct vertices $(x_1,y_1),(x_2,y_2)\in V(G_1)\times V(G_2)$ are adjacent in $G_1⊠G_2$ if and only if $x_1=x_2$ and $y_1{y}_2\in E(G_2)$ or $y_1=y_2$ and $x_1{x}_2\in E(G_1)$ or $x_1{x}_2\in E(G_1)$ and $y_1{y}_2\in E(G_2)$. In this paper, we give the $g(\le 3)$-$extra$ $connectivity$ of $G_1⊠G_2$, where $G_i$ is a maximally connected $k_i(\ge 2)$-regular graph for $i=1,2$. As a byproduct, we get $g(\le 3)$-$extra$ conditional fault-diagnosability of $G_1⊠G_2$ under $PMC$ model.