In ISGCI, unit interval graphs are denoted as ($C_{n+4}$,$S_3$,claw,net)-free. Is this an accurate notation?
You can look up the small graphs that ISGCI uses to define classes. Specifically, $S_3$ is the following "sun" graph:



ISGCI defines sun graphs as


  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 $W$&nbsp;is independent and $u_i$&nbsp;is adjacent to&nbsp;$w_j$ iff $i=j$ or $i=j+1\pmod{n}$.


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