of degree 2 is adjacent to $d-2$ 1-vertices, as depicted in Figures \[fig:Y\]–\[fig:Phi\]. This suffices to address all other instances of these configurations that may occur.

\[claim:red\_cat\] The following caterpillars with maximum degree $d$ are reducible:

1.  (Borodin and Ivanova [@BI]) For $d = 3$, the $8$-caterpillar is 5-reducible.

2.  (Wang and Zhao [@WZ]) For $d \geq 4$, the $(2d-2)$-caterpillar is $(2d-1)$-reducible.

These caterpillars are likely the smallest that are reducible for each degree $d$. Thus, the bounds in Theorems \[thm:bi\] and \[thm:wz\] are best possible using only Lemma \[lma:nrs\]. To improve these bounds, we demonstrate larger reducible configurations and use a more complicated discharging argument.

\[clm:Reducible\] The following configurations with maximum degree 3 are 5-reducible:

1.  $Y(1,6,7)$, $Y(2,5,6)$ and $Y(3,4,5)$.

2.  $H(7,7;0;3,7),\, H(7,7;0;4,6),\, H(7,7;0;5,5),\, H(6,7;0;3,7),\, H(6,7;0;4,6),\\
            H(6,7;0;5,5),\, H(6,6;1;2,7),\, H(6,6;1;3,6),\, H(6,6;1;4,5),\, H(5,7;1;2,7),\\
            H(5,7;1;3,6),\, H(5,7;1;4,5),\, H(4,7;2;1,7),\, H(4,7;2;2,6),\, H(4,7;2;3,5),\\
            H(4,7;2;4,4),\, H(3,7;3;1,6),\, H(3,7;3;2,5) \text{ and }  H(3,7;3;3,4).$

3.  $\Phi(7,0,7,1),\, \Phi(7,0,6,1),\, \Phi(6,0,7,1),\, \Phi(6,1,6,1),\, \Phi(7,1,5,1), \Phi(5,1,7,1),\\
             \Phi(7,2,4,1),\, \Phi(4,2,7,1),\, \Phi(7,3,3,1),\, \Phi(3,3,7,1) \text{ and }\Phi(3,7,0,7).$

\[clm:Reducible4\] The following configurations with maximum degree 4 are $7$-reducible: $$Y(2,4,4),\ Y(1,5,5),\ Y(2,4,5),\ Y(3,4,4),\text{ and }Y(2,5,5).$$

Proof of Theorem \[thm:sparse\]
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Among graphs $G$ with $\operatorname{mad}(G) < 2 + \frac{1}{7}$ not containing $S_3$, $S_4$, or $S_7$, with ${\chi_{s}'(G)}> 5$, select $G$ while 