Havel-Thomas-Walls graph* is any graph obtained from such a graph $G'$ by patching, and $u_1u_3$ is its *interface pair*.

![Graphs related to Theorem \[thm-crtri\].[]{data-label="fig-havel"}](fig-havel "fig:") ![Graphs related to Theorem \[thm-crtri\].[]{data-label="fig-havel"}](fig-havel "fig:") ![Graphs related to Theorem \[thm-crtri\].[]{data-label="fig-havel"}](fig-havel "fig:")

Let a *tent* be a graph embedded in the disk with the ring $v_1v_2v_3v_4$, containing vertices $z_1$ adjacent to $v_1$ and $v_2$, and $z_2$ adjacent to $v_3$ and $v_4$, such that all faces other than $v_1v_2z_1$ and $v_3v_4z_2$ have length $4$, and $v_1v_3,v_2v_4\not\in E(G)$, see Figure \[fig-havel\](c).

\[thm-crtri\] Let $G$ be a graph embedded in the disk with at most two triangles and with the ring of length $4$. If $G$ is critical, then $G$ is either a tent, or obtained from a patched Havel-Thomas-Walls graph by framing on its interface pair.

It is important to note that the precolorings of the rings which extend to graphs appearing in the conclusions of Theorems \[thm-many\] and \[thm-crtri\] can be precisely described, as we show in Section \[sec-htw\]. Therefore, in the structure theorem we aim for, it is satisfactory to have pieces that are patched reduced Thomas-Walls graphs. Furthermore, in Theorem \[thm-many\], all faces $f$ of $H$ distinct from the rings are bounded by contractible 