Does graph G with all vertices of degree 3 have a cut vertex? I'm asked to draw a simple connected graph, if possible, in which every vertex has degree 3 and has a cut vertex. I tried drawing a cycle graph, in which all the degrees are 2, and it seems there is no cut vertex there. I know, so far, that, by the handshaking theorem, the number of vertices have to be even and they have to be greater than or equal to 4. So, I kept drawing such graphs but couldn't find one with a cut vertex. I have a feeling that there must be at least one vertex of degree one but I don't know how to formally prove this, if its true. I'd appreciate if someone can help with that.
Not necessarily true, for example complete graph of 4 vertices have no cut vertex. But there exists a graph G with all vertices of degree 3 and there
is a cut vertex. See the picture. Red vertex is the cut vertex.