Let R be a ring with unity. The cozero-divisor graph of a ring R is an
undirected simple graph whose vertices are the set of all non-zero and non-unit
elements of R and two distinct vertices x and y are adjacent if and only
if xβ/Ry and yβ/Rx. The reduced cozero-divisor graph of a ring
R, is an undirected simple graph whose vertex set is the set of all
nontrivial principal ideals of R and two distinct vertices (a) and (b)
are adjacent if and only if (a)ξ β(b) and (b)ξ β(a). In
this paper, we characterize all classes of finite non-local commutative rings
for which the cozero-divisor graph and reduced cozero-divisor graph is of genus
two.Comment: 16 Figure