# In ISGCI, unit interval graphs are denoted as ($C_{n+4}$,$S_3$,claw,net)-free. Is this an accurate notation?

When I search unit interval graphs in ISGCI, it says that the unit interval graphs (UIG) are equivalent to ($$C_{n+4}$$,$$S_3$$,claw,net)-free graphs.

I am confused about the definition of an $$S_3$$ graph. I am aware that in some resources, $$S_n$$ means $$K_{1,n}$$, and in others $$S_{n-1}$$ means $$K_{1,n}$$.

However, in both cases, ($$C_{n+4}$$,$$S_3$$,claw,net)-free $$\equiv$$ UIG seems like a wrong notation.

If $$S_3$$ is $$K_{1,3}$$, then $$S_3$$ is also a claw graph. Thus, ($$S_3$$,claw)-free is a redundant notation. On the other hand, if $$S_3$$ is $$K_{1,2}$$, then it is wrong because a path of length 3 is realizable as a unit interval graph.

If this notation is true, then can I also write UIG $$\equiv$$ ($$C_{n+4}$$,$$S_3$$,$$K_{1,2}$$,$$P_2$$,claw,net)-free?
If not, then is it correct to write that UIG $$\equiv$$ ($$C_{n+4}$$,claw,net)-free?

You can look up the small graphs that ISGCI uses to define classes. Specifically, $$S_3$$ is the following "sun" graph:

a chordal graph on $$2n$$ nodes ($$n\geq3$$) whose vertex set can be partitioned into $$W = \{w_1,\dots,w_n\}$$ and $$U = \{u_1, \dots,u_n\}$$ such that $$W$$ is independent and $$u_i$$ is adjacent to $$w_j$$ iff $$i=j$$ or $$i=j+1\pmod{n}$$.

Implicitly, the subgraph induced by $$U$$ can be anything. Wolfram Mathworld also talks about "complete sun graphs", which is the special case where $$U$$ is a clique.