A long exact sequence of path homology of digraphs

Abstract In this paper, we develop a long exact sequence for the path homology of digraphs, providing a useful tool for computing the path homology of digraphs. One application of this result is the proof of a conjecture proposed by S. Chowdhury, which was initially observed through extensive computational experiments. Another interesting application demonstrates that the path homology of n -dimensional grid-like digraphs is concentrated in dimension ≤ n - 1 {\leq n-1} .