We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix
$\Gcaron_ \Delta \isin \mathbb{M}_n(\mathbb{Z})$
and the Coxeter transformation
$\Phi_A : \mathbb{Z}^n \rarr \mathbb{Z}^n$
defined by a quasi-triangular matrix morsification
$A \isin \mathbb{M}_n(\mathbb{Z})$
of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure
$\Gamma(\Rscr_\Delta, \Phi_A)$
on the finite set of all ΦA-orbits of the irreducible root system
$\Rscr_\Delta := {v \isin \mathbb{Z}^n; v \cdot \Gcaron_\Delta \cdot v^{tr} = 1}$
. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure
$\Gamma(\Rscr_I, \Phi_I)$
on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form
$\qIcirc : \mathbb{Z}_I \rarr \mathbb{Z}$
.
