Abstract
A graph G is s -Hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a Hamiltonian-connected graph. In this paper, for positive integer s, a graph G is s-trail-connected if for any vertex subset X with |X|≤ s, G − X is trail-connected; a graph G is s-edge-trail-connected if for any edge subset Y with|Y|≤ s, G \\ Y is trailconnected. The main purpose of this paper is to depict the structure of 1-trail-connected and 1-edge-trail-connected square tree.
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