Corrigendum to "The path-partition problem in block graphs"

Recently, Wong [1] pointed out that Yan and Chang’s [2] linear-time algorithm for the path-partition problem for block graphs is not correct, by giving the following example. Suppose G is the graph consisting of a vertex w and a set of triangles {xi, yi, zi} such that each xi is adjacent to w for 1 i k, where k 3. Then p(G) = k − 1, but Yan and Chang’s algorithm gives p(G)= 1. He also traced the algorithm for the graph in Fig. 2 of [2] in a different ordering to get an inconsistent value. He then gave a linear-time algorithm for the problem. We clarify two things. First, Yan and Chang’s algorithm is correct except for a typo: the J should be J ∗ in line 18 of Algorithm PPN. This is because it applies Theorem 3 for the graph G′, the composition