Timeline for Characterizing 1-ended graphs

10 events

when toggle format	what		by	license	comment
Jan 14, 2018 at 9:39	vote	accept	I. Haage
Jan 13, 2018 at 21:52	comment	added	Taras Banakh		The implication 2 $\Rightarrow$ 1 also is simple: just observe that for any finite connected subgraph $\Gamma$ of a 1-ended graph $G$ the complement $G\setminus\Gamma$ has finitely many connected components and exactly one of them is infinite. Using this fact, by induction, one can easily construct the required infinite path in $G$.
Jan 13, 2018 at 21:45	answer	added	Lee Mosher		timeline score: 4
Jan 13, 2018 at 20:28	history	edited	YCor		edited tags
Jan 13, 2018 at 20:26	comment	added	YCor		Maybe you could mention that clearly 1 implies 2, so you want to know about the converse.
Jan 13, 2018 at 19:57	comment	added	I. Haage		Thanks for pointing this out, I edited to add the assumption of connectedness.
Jan 13, 2018 at 19:57	history	edited	I. Haage	CC BY-SA 3.0	added 10 characters in body
Jan 13, 2018 at 19:51	comment	added	Arun Debray		No. Let $G$ be a graph with a vertex for every natural number and an edge between $n$ and $n+1$ for all $n$, and let $G_+$ denote $G$ together with an additional, isolated vertex. Then, $G_+$ is 1-ended, but there's no path that reaches all vertices of $G_+$.
Jan 13, 2018 at 19:46	review	First posts
Jan 13, 2018 at 19:52
Jan 13, 2018 at 19:42	history	asked	I. Haage	CC BY-SA 3.0