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A finite, undirected graph is called locally connected , if the neighborhood of every vertex induces a connected subgraph. In this paper we study the existence of edges in locally connected k -connected graphs whose contraction keeps the graph locally connected k -connected. As an application, we...
We determine when the topological spaces | G | naturally associated with a graph G and its ends are metrizable or compact. In the most natural topology, | G | is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone's theorem that metric spaces are...