Abstract
In the first one in this series of two papers, we have proved that every 3-connected { K 1 , 3 , N 1 , 3 , 3 } $\{{K}_{1,3},{N}_{1,3,3}\}$ -free graph is Hamilton-connected. In this paper, we continue in this direction by proving that every 3-connected { K 1 , 3 , X } $\{{K}_{1,3},X\}$ -free graph, where X ∈ { N 1 , 1 , 5 , N 2 , 2 , 3 } $X\in \{{N}_{1,1,5},{N}_{2,2,3}\}$ , is Hamilton-connected (where N i , j , k ${N}_{i,j,k}$ is the graph obtained by attaching endvertices of three paths of lengths i , j , k $i,j,k$ 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 X $X$ such that every 3-connected { K 1 , 3 , X } $\{{K}_{1,3},X\}$ -free graph is Hamilton-connected.
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