Robust Connectivity of Graphs on Surfaces

Abstract

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