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 $T(Gamma_{I} (R))$. It is the (undirected) graph with elements of $R$ as vertices, and for distinct $x, y in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y in P(I)$. The properties and possible structures of the two (induced) subgraphs $P(Gamma_{I} (R))$ and $bar {P}(Gamma_{I} (R))$ of $T(Gamma_{I} (R))$, with vertices $P(I)$ and $R - P(I)$, respectively are studied.