  We consider an extension of the triangular-distance Delaunay graphs
(TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex
distance is defined by a fixed-oriented equilateral triangle $\triangledown$,
and there is an edge between two points in $P$ if and only if there is an empty
homothet of $\triangledown$ having the two points on its boundary. We consider
higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains
an edge between two points if the interior of the homothet of $\triangledown$
having the two points on its boundary contains at most $k$ points of $P$. We
consider the connectivity, Hamiltonicity and perfect-matching admissibility of
$k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.
