On Super Connectedness and Super Restricted Edge-Connectedness of Total Graphs

Abstract

A graph $G$ is $super-connected$, or $super-\text{(}\kappa \text{)}$, if every minimum vertex-cut isolates a vertex of $G$. Similarly, $G$ is $super-restrictededge-connected$, or $super-\text{(}{\lambda }^{\prime }\text{)}$, if every minimum restricted edge-cut isolates an edge. We consider the total graph $T(G)$ of $G$, which is formed by combining the disjoint union of $G$ and the line graph $L(G)$ with the lines of the subdivision graph $S(G)$; for each line $l=(u,v)$ in $G$, there are two lines in $S(G)$, namely $(l,u)$ and $(l,v)$. In this paper, we prove that $T(G)$ is super-$\kappa$ if $G$ is super-$\kappa$ graph with $\delta (G)\ge 4$. $T(G)$ is super-${\lambda }^{\prime }$ if $G$ is $k$-regular with $\kappa (G)\ge 3$. Furthermore, we provide examples demonstrating that these results are best possible.