Completion problems of partial $N_0^1$-matrices under directed 2-trees

A matrix completion problem asks whether a partial matrix has a completion to a conventional matrix with a desired property. C. Mendes Ara $\acute{u}$jo and J. R. Torregrosa explored the completion problem of a combinatorially symmetric N0-matrix by applying an undirected graph. However, in practical applications such as seismic data reconstruction, data transmission, and engineering computation data are often incomplete and must be represented by a non-combinatorially symmetric matrix. In this paper, we discuss the completion problem of a non-combinatorially symmetric partial matrix by using a directed graph and prove that a non-combinatorially symmetric partial matrix under a directed 2-tree is completed as an $N_0^1$-matrix.