The total graph of a commutative semiring with respect to proper ideals

Let I be a proper ideal of a commutative semiring R and let P(I) be the set of all elements of R that are not prime to I. In this paper, we investigate the total graph of R with respect to I, denoted by T(ΓI(R)). It is the (undirected) graph with elements of R as vertices, and for distinct x,y∈R, the vertices x and y are adjacent if and only if x∈P(I). The properties and possible structures of the two (induced) subgraphs P(ΓI(R)) and P¯(ΓI(R)) of T(ΓI(R)), with vertices P(I) and R−P(I), respectively are studied.