Let $\ X:=M^n\ $ be a connected manifold (with or without boundary) of dimension $\ n\ge 1,\ $ or let $\ X:=S^1\ $ be a circle. Then

$(E)\qquad X\setminus\{p\}\ $ is connected for every $\ p\in X$

Only other hand, of the mentioned spaces only the circle has the stronger property

$(B)\qquad X\setminus Y\ $ is connected for every connected subspace $\ Y\subset X$.

Thus $\ (B)\Rightarrow (E).\ $ Now, given a naturally or classically defined class of topological spaces which already have property $(E)$ one would like to prove property $(B)$. In particular, let me propose the following two conjectures (and their unspoken about obvious variations; the first conjecture is most likely already know, in which case I would like to ask about references):

1:   Let $\ X\ $ be an arbitrary connected topological graph (the body of any finite 1-dimensional simplicial complex) which has no end-points (i.e. $(E)$ holds). Does $\ X\ $ have property $(B)$?

2:   Let $\ X\ $ be an arbitrary connected 1-dimensional topological space, which has property $(E)$. Does $\ X\ $ have property $(B)$?

Here one may consider one of the three classical topological dimensions: $\dim$, ind, or Ind. One may consider separable metric spaces (when the three dimensions are equivalent), or Hausdorf compact spaces for the covering dimension $\dim$. Other variations are possible, interesting and welcome.

REMARK 1   For every Hausdorff connected compact space $\ X,\ $ which has at least two different points, there are at least two different points $\ p\in X\ $ such that $\ X\setminus\{p\}\ $ is connected.

REMARK 2   Property $(B)$ discussed in this post is antipodal to the notion of the biconnected spaces.