  A tromino tiling problem is a packing puzzle where we are given a region of
connected lattice squares and we want to decide whether there exists a tiling
of the region using trominoes with the shape of an L. In this work we study a
slight variation of the tromino tiling problem where some positions of the
region have pegs and each tromino comes with a hole that can only be placed on
top of the pegs. We present a characterization of this tiling problem with pegs
using flow networks and show that (i) there exists a linear-time parsimonious
reduction to the maximum-flow problem, and (ii) counting the number of such
tilings can be done in linear-time. The proofs of both results contain
algorithms that can then be used to decide the tiling of a region with pegs in
$O(n)$ time.
