Summary: Construct a graph as follows. Take a circle, and a collection of intervals from it, no three of which have union the entire circle; take a finite set of points $V$ from the circle; and make a graph with vertex set $V$ in which two vertices are adjacent if they both belong to one of the intervals. Such graphs are “long circular interval graphs,” and they form an important subclass of the class of all claw-free graphs. In this paper we characterize them by excluded induced subgraphs. This is a step towards the main goal of this series, to find a structural characterization of all claw-free graphs. This paper also gives an analysis of the connected claw-free graphs $G$ with a clique the deletion of which disconnects $G$ into two parts both with at least two vertices. [For the other parts see: I, ibid. 97, No.\,6, 867‒903 (2007; Zbl 1128.05031); II, ibid. 98, No.\,2, 249‒290 (2008; Zbl 1137.05040); IV, ibid. 98, No.\,5, 839‒938 (Zbl 1152.05038); V, ibid. 98, No.\,6, 1173‒1410 (2008; Zbl 1196.05043).]