Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.

More precisely, let $h:X\to X$ be a homeomorphism. If $U$ is a component of $\mathbb R^2\setminus X$ and $B=\partial U$, does there exist a component $V$ of $\mathbb R^2\setminus X$ such that $h[B]=\partial V$?

I am mostly interested in the case when $X$ is locally connected and $1$-dimensional. Under these assumptions, $B$ is a locally connected continuum, and is essentially a simple closed curve.

Let $X\subset \mathbb R^2$ be compact and connected. My question is whether homeomorphisms of $X$ preserve boundaries of complementary components.

More precisely, let $h:X\to X$ be a homeomorphism.

Question 1. If $U$ is a component of $\mathbb R^2\setminus X$ and $B=\partial U$, does there exist a component $V$ of $\mathbb R^2\setminus X$ such that $h[B]=\partial V$?

I am mostly interested in the case when $X$ is locally connected and $1$-dimensional. Under these assumptions, $B$ is a locally connected continuum, and is essentially a simple closed curve.

EDIT: Question 1 has a negative answer even for topological graphs, as pointed out in the comments. Maybe the better question is:

Question 2. Let $B$ be the union of all boundaries of complementary components of $X$. Then for every homeomorphism $h$ of $X$, $h[B]=B$.

Note that in the case that $X$ is locally connected, $B$ is equivalent to the set of accessible points of $X$ (i.e. points which are arcwise accessible from $\mathbb R^2\setminus X$).