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 $strong$ $product$ $G_1 \boxtimes G_2$ of graphs $G_1$ and $G_2$ is the graph with vertex set $V(G_1 \boxtimes G_2)=V(G_1)\times V(G_2)$, where two distinct vertices $(x_{1}, y_{1}),(x_{2}, y_{2}) \in V(G_1)\times V(G_2)$ are adjacent in $G_1 \boxtimes G_2$ 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 $g\ (\leq 3)$-$extra$ $connectivity$ of $G_1\boxtimes G_2$, where $G_i$ is a maximally connected $k_i\ (\geq 2)$-regular graph for $i=1,2$. As a byproduct, we get $g\ (\leq 3)$-$extra$ conditional fault-diagnosability of $G_1\boxtimes G_2$ under $PMC$ model

The <i>g</i>-Extra Connectivity of the Strong Product of Paths and Cycles

Let G be a connected graph and g be a non-negative integer. A vertex set S of graph 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 is the minimum cardinality of a g-extra cut of G if G has at least one g-extra cut. For two graphs G1=(V1,E1) and G2=(V2,E2), the strong product G1⊠G2 is defined as follows: its vertex set is V1×V2 and its edge set is {(x1,x2)(y1,y2)|x1=x2 and y1y2∈E2; or y1=y2 and x1x2∈E1; or x1x2∈E1 and y1y2∈E2}, where (x1,x2),(y1,y2)∈V1×V2. 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