Robust Connectivity of Graphs on Surfaces

Let \(\varLambda (T)\) denote the set of leaves in a tree T. One natural problem is to look for spanning T given graph G such that as large possible. Recently, similar but stronger notion called robust connectivity was introduced, which defined minimum value \(\frac{|R \cap \varLambda (T)|}{|R|}\) taken over all nonempty subsets \({R\subseteq V(G)}\), where \(T = T(R)\) on chosen maximize \({|R (T)|}\). We prove tight asymptotic bound \(\varOmega (\gamma ^{-\frac{1}{r}})\) r-connected graphs Euler genus \(\gamma \). Moreover, we give surprising connection between with an edge-maximal embedding surface and surface, describes what extent induced subgraphs embedded can be cut out from without splitting into multiple parts. For planar graphs, this provides equivalent formulation long-standing conjecture Albertson Berman.