On the number of points skipped by a transient (1,2) random walk on the line

Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the criteria for the finiteness of the number of cutpoingts of the nearest neighbor random walk on the line by Cs'aki, F"olders, R'ev'esz [J Theor Probab (2010) 23: 624-638].