It is somewhat surprising (to me) that what to me seems the simplest nontrivial example of theorems exactly fitting the question in the OP has not yet been mentioned in this thread: the embeddings of the Möbius ladders, which are finite simple undirected graphs, into $\mathbb{R}^3$.
This is an answer to
whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) embeddings of graphs into R^3 or S^3. If yes, can anyone show me any reference?
at least in the sense that in the very interesting article Erica Flapan: The Symmetries of the Möbius Ladder. Math. Ann. 283, 271-283 (1989), which I thinkg, could be given a fruitful revival from a point of view of constructive mathematics (i.e., how much of Flapan's proof/theorems can be done constructively?), the following was done
in Section 1 of loc. cit. a proof is given that there exists a graph $G$ and an automorphism $\alpha$ of $G$ as an abstract graph, such that there does not exist any embedding of (the geometric realization $\lvert G\rvert$ of ) $G$ into $\mathbb{R}^3$ such that $\alpha$ could be realized by an element of the group of leaving-$\lvert G\rvert$-invariant-as-a-point-set diffeomorphisms of $\mathbb{R}^3$. Note that in this rendition rendered the author's "group of homeomorphisms of $G$ up to isotopy" into what to me evidently seems equivalent and clearer "automorphism of $G$ as an abstract graph". The example graph used by the autor is $G:=$complete graph with six vertices.
in Section 2 of loc. cit. first a proof in classical logic is given that for any embedding $\eta\colon M_r\to S^3$ of the $r$-rung Möbius ladder $M_r$ into the $3$-sphere $\mathbb{S}^3$, any orientation-reversing diffeomorphism $\varphi\colon S^3\to S^3$ has the property that if it fixes $\mathrm{im}(\eta)$ as a pointset, then $\varphi$ does not fix at least $r-2$ of the $r$ rungs of $M_r$. This is, partly,, an interesting counterfactural (since the author later gives a proof that for an odd number of rungs, such $\varphi$ are impossible): roughly, if there is a symmetry of the Euclidean-space-embedded Möbius ladder, then it must needs jumble almost all the ladder's rungs. This implication is then put to use to give a proof that for an odd number of $r$ungs, such a $\varphi$ is impossible. Roughly, you cannot reverse the orientation of an embedded odd-rung-number Möbius ladder graph by a Euclidean symmetry. The author then gives an example that for even number of rungs, such isometries do exist. I perceived this to be a result which is very relevant to the OP; it in particular is an interaction between a combinatorial property of the abstract graph (number of rungs) and a a mainstay of knot theory: slightly vaguely, one could say: Flapan gave a proof that even-rung Möbius ladders are amphichiral, while odd-rung Möbius ladder are non-amphichiral.