A graph G is
$$\{X,Y\}$$
-free if it contains neither X nor Y as an induced subgraph. Pairs of connected graphs X,Y such that every 3-connected
$$\{X,Y\}$$
-free graph is Hamilton-connected have been investigated recently in (2002, 2000, 2012). In this paper, it is shown that every 3-connected
$$\{K_{1,3},N_{1,2,3}\}$$
-free graph is Hamilton-connected, where
$$N_{1,2,3}$$
is the graph obtained by identifying end vertices of three disjoint paths of lengths 1,2,3 to the vertices of a triangle.
