Hamilton-connected {claw,net}-free graphs, II

In the first one in this series of two papers, we have proved that every 3-connected (Formula presented.) -free graph is Hamilton-connected. In this paper, we continue in this direction by proving that every 3-connected (Formula presented.) -free graph, where (Formula presented.), is Hamilton-connected (where (Formula presented.) is the graph obtained by attaching endvertices of three paths of lengths (Formula presented.) to a triangle). This together with a previous result of other authors completes the characterization of forbidden induced generalized nets implying Hamilton-connectedness of a 3-connected claw-free graph. We also discuss remaining open cases in a full characterization of connected graphs (Formula presented.) such that every 3-connected (Formula presented.) -free graph is Hamilton-connected.