The Restricted Edge-Connectivity of Strong Product Graphs (2401.15549v1)

Abstract: 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.