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.