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

Abstract

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.