What is $MFIS(X)$ if $X = \{ G : \Delta(G) \leq 2 \}$?

Question

What is the set of minimal forbidden induced subgraphs of the class X of graphs with maximum vertex degree 2? I know that X is the set of all chordless cycles and paths, and my intuition was that X was the set of claw-free graphs as the claw $(K_{1,3})$ contains a vertex of degree 3.

Answer 1

Forbidding only the induced subgraph $K_{1,3}$ doesn't rule out examples such as the complete graph $K_n$ for large $n$.

Whenever you have a vertex $v$ with $\deg(v) = 3$, you have $4$ vertices $v, w_1, w_2, w_3$ where edges $vw_1, vw_2, vw_3$ are present. Then $\{v, w_1, w_2, w_3\}$ induce one of several subgraphs, depending on which of the edges $w_1w_2, w_1w_3, w_2w_3$ you've also got. Forbid all of them, and you're golden.