
On the Average (Edge)Connectivity of Minimally k(Edge)Connected Graphs
Let G be a graph of order n and let u,v be vertices of G. Let κ_G(u,v) d...
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Connected max cut is polynomial for graphs without K_5 e as a minor
Given a graph G=(V, E), a connected cut δ (U) is the set of edges of E l...
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Matchings in 1planar graphs with large minimum degree
In 1979, Nishizeki and Baybars showed that every planar graph with minim...
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A Robust Advantaged Node Placement Strategy for Sparse Network Graphs
Establishing robust connectivity in heterogeneous networks (HetNets) is ...
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Smaller extended formulations for spanning tree polytopes in minorclosed classes and beyond
Let G be a connected nvertex graph in a proper minorclosed class 𝒢. We...
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On the dichromatic number of surfaces
In this paper, we give bounds on the dichromatic number χ⃗(Σ) of a surfa...
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Robustness: a New Form of Heredity Motivated by Dynamic Networks
We investigate a special case of hereditary property in graphs, referred...
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Robust Connectivity of Graphs on Surfaces
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 wellknown NPhard problem. Throughout recent decades, this problem has 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 κ_ρ(G), defined as the minimum value R ∩Λ (T)/R taken 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 nonregular 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 rconnected graphs of Euler genus γ. Moreover, we give a surprising connection between the robust connectivity of graphs with an edgemaximal 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 longstanding 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.
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