2024/04/26 by Aurichi, Leandro Fiorini, Júnior, Paulo Magalhães, Real, Lucas · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2404.17116
The notion of ends in an infinite graph G might be modified if we consider them as equivalence classes of infinitely edge-connected rays, rather than equivalence classes of infinitely (vertex-)connected ones. This alternative definition yields to the edge-end space ΩE(G) of G, in which we can endow a natural (edge-)end topology. For every graph G, this paper proves that ΩE(G) is homeomorphic to Ω(H) for some possibly another graph H, where Ω(H) denotes its usual end space. However, we also show that the converse statement does not hold: there is a graph H such that Ω(H) is not homeomorphic to ΩE(G) for any other graph G. In other words, as a main result, we conclude that the class of topological spaces ΩE = \ΩE(G) : G graph\ is strictly contained in Ω= \Ω(H) : H graph\.