  In 1988, I. Beck introduced the notion of a zero-divisor graph of a
commutative rings with $1$. There have been several generalizations in recent
years. In particular, in 2007 J. Coykendall and J. Maney developed the
irreducible divisor graph. Much work has been done on generalized
factorization, especially $\tau$-factorization. The goal of this paper is to
synthesize the notions of $\tau$-factorization and irreducible divisor graphs
in domains. We will define a $\tau$-irreducible divisor graph for non-zero
non-unit elements of a domain. We show that by studying $\tau$-irreducible
divisor graphs, we find equivalent characterizations of several finite
$\tau$-factorization properties.
