  Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with
respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An
HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph
$H$, a graph $G$ is $H$-free if $G$ has no induced subgraph isomorphic to $H$.
The main motivation for this paper originates from a theorem of Duffus, Gould,
and Jacobson (1981), which characterizes all the minimal connected graphs with
no Hamiltonian path. In 1998, Brousek characterized all the claw-free
HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all
(not only the minimal) 2-connected non-Hamiltonian $\{K_{1,3},
N_{3,1,1}\}$-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022),
characterized all triangle-free HC-obstructions and all the HC-obstructions
which are split graphs. A wheel is a graph obtained from a cycle by adding a
new vertex with at least three neighbors in the cycle. In this paper we
characterize all the HC-obstructions which are wheel-free graphs.
