In this paper, we intend to show that under not too restrictive conditions, results much stronger than the one obtained earlier by Hejduk could be established in category bases.
In this article we introduce the zero-divisor graphs $Gamma_mathscr{P}(X)$ and $Gamma^mathscr{P}_infty(X)$ of the two rings $C_mathscr{P}(X)$ and $C^mathscr{P}_infty(X)$; here $mathscr{P}$ is an ideal of closed sets in $X$ and $C_mathscr{P}(X)$ is th
e aggregate of those functions in $C(X)$, whose support lie on $mathscr{P}$. $C^mathscr{P}_infty(X)$ is the $mathscr{P}$ analogue of the ring $C_infty (X)$. We find out conditions on the topology on $X$, under-which $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) becomes triangulated/ hypertriangulated. We realize that $Gamma_mathscr{P}(X)$ (respectively, $Gamma^mathscr{P}_infty(X)$) is a complemented graph if and only if the space of minimal prime ideals in $C_mathscr{P}(X)$ (respectively $Gamma^mathscr{P}_infty(X)$) is compact. This places a special case of this result with the choice $mathscr{P}equiv$ the ideals of closed sets in $X$, obtained by Azarpanah and Motamedi in cite{Azarpanah} on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals $mathscr{P}$ and $mathscr{Q}$ on $X$ and $Y$ respectively that the rings $C_mathscr{P}(X)$ and $C_mathscr{Q}(Y)$ are isomorphic if and only if $Gamma_mathscr{P}(X)$ and $Gamma_mathscr{Q}(Y)$ are isomorphic.
