Robust Connectivity of Graphs on Surfaces

Abstract

Let Λ(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 Λ(T ) is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Equivalently, the same problem can be formulated as the minimum connected dominating set problem, where the task is to find a smallest subset of vertices D ⊆ V (G) such that every vertex of G is in the closed neighborhood of D. Throughout recent decades, these two equivalent problems have received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks.Recently, a similar but stronger notion was defined by Bradshaw, Masařík, and Stacho [Flexible List Colorings in Graphs with Special Degeneracy Conditions, ISAAC 2020]. They introduced a new invariant for a graph G, called the robust connectivity and written κρ(G), defined as the minimum valuetaken over all nonempty subsets R ⊆ V (G), where T = T (R) is a spanning tree on G chosen to maximize |R ∩ Λ(T )|. Large robust connectivity was originally used to show flexible choosability in non-regular graphs.In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of Ω(γ − 1 r ) for the robust connectivity of r-connected graphs of Euler genus γ. 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 [A conjecture on planar graphs, 1979], which states that every planar graph on n vertices contains an induced forest of size at least n/2.