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

Abstract

A graph \(G\) is \({super-connected}\), or \({super-\(\kappa\)}\), if every minimum vertex-cut isolates a vertex of \(G\). Similarly, \(G\) is \({super-restricted \;edge-connected}\), or \({super-\(\lambda’\)}\), 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 \(\delta(G) \geq 4\). \(T(G)\) is super-\(\lambda’\) if \(G\) is \(k\)-regular with \(\kappa(G) \geq 3\). Furthermore, we provide examples demonstrating that these results are best possible.