Comparison of symbolic and ordinary powers of parity binomial edge ideals

In this paper, we investigate when symbolic and ordinary powers of the parity binomial edge ideal of a graph fail to be equal. It turns out that if IG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}_{G}$$\end{document} is the parity binomial edge ideal of a graph G, then in each of the following cases the symbolic power IG(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}_{G}^{(t)}$$\end{document} and the ordinary power IGt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {I}}_{G}^t$$\end{document} are not equal for some t: (i) the clique number of G is greater than 3; (ii) G has a net; or (iii) G has a PT as an induced subgraph.