Does this paper imply graph isomorphism is polynomial for cubic and $4$-regular graphs? This paper gives example
of polynomial GI for certain graphs.

Probably I am misunderstanding the paper, but appears
to me it implies polynomial GI for cubic, $4$-regular
and probably higher degree regular graphs.

GI for regular graphs is GI complete.

On p. 7 the graphs $H(a,b,c)$ are defined. They are a claw with
$a+b$ leaves, $a$ edges are subdivided and $c$ isolated
vertices are added.

On p. 8 Theorem 4. Isomorphism of $(H(0, b, c), \overline{H(0, b' , c' ))}$-free graphs is in $P$ when:

(2.) $c, c′ \le 1$ and $b, b' \ge 1$.

Take $c=c'=1,b=b'=5$.

$H=H(0,5,1)$ is $K_{1,5} + K_1$.

Both $H$ and $\overline{H}$ have degree $5$ vertex,
so neither can be induced subgraph of cubic or
$4$ regular graph.

By taking larger $b$ this works for higher
degree regular graphs.


Does the paper imply GI is in $P$ for cubic
and $4$-regular graphs?
Does the paper imply GI is in $P$ for higher
degree regular graphs (the running time might
depend on $b$, not sure)?
Their paper doesn't say anything about general regular graphs. Note that the $b$ is constant. Actually all of their excluded subgraphs are constant. Their polynomial time algorithm is consist of considering all possible $K_{2b+1}$'s in that case. This causes to running time $\Omega(n^{2b+1})$. Means if we want to use their algorithm for $r$-regular graph then we should use running time $\Omega(n^{2r+1})$, means when $r$ is not fix their algorithm is not polynomial time in the size of input. Actually we can say that they proved the problem in that case belongs to XP when the problem parametrized by $b$. Don't mix parametrized complexity with usual polynomial time algorithms.