A class of "minimally 2-vertex-connected graphs" - that is, 2-vertex-connected graphs which have the property that removing any one vertex (and all incident edges) renders the graph no longer 2-connected - have come up in my research.


Dirac wrote a paper on "minimally 2-connected graphs" (G. A. Dirac, Minimally 2-connected graphs, J. Reine Angew. Math. 228 (1967),. 204-216), which gives quite a detailed description of the structure of such graphs. However, in his sense, minimal 2-connectivity means that deleting any EDGE leaves a graph which is not 2-connected, which is not an equivalent property to the vertex-deletion one. Does anyone know anything about graphs with the latter property?


In the hope of stimulating some discussion, here is a wildly speculative and vague conjecture: The only graphs satisfying this property are simple cycles, and certain cycles with chords.



Minimally 2-vertex-connected graphs?
Here is a more general family:


Draw your favorite tree in the plane, with circles for the nodes and "thick" lines for the edges. Now turn every circle into a cycle, and every thick line into a pair of parallel paths $p\_1, \ldots, p\_m$ and $q\_1, \ldots, q\_n$ with various crossbraces. The crossbraces just have to follow the rule that if $p\_i$ is connected to $q\_\ell$ and $p\_j$ is connected to $q\_k$, for $i&lt;j$ and $k&lt;\ell$, then $j =i+1$ or $\ell=k+1$.


This is probably still not close to a complete characterization, but at least shows that the class is a lot broader than the small class you posited to promote discussion.


