Gallai's question and constructions of almost hypotraceable graphs

Consider a graph G in which the longest path has order |V(G)|−1. We denote the number of vertices v in G such that G−v is non-traceable with tG. Gallai asked in 1966 whether, in a connected graph, the intersection of all longest paths is non-empty. Walther showed that, in general, this is not true. In a graph G in which the longest path has |V(G)|−1 vertices, the answer to Gallai's question is positive iff tG≠0. In this article we study almost hypotraceable graphs, which constitute the extremal case tG=1. We give structural properties of these graphs, establish construction methods for connectivities 1 through 4, show that there exists a cubic 3-connected such graph of order 28, and draw connections to works of Thomassen and Gargano et al.