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analysis: **Faisal** **Nadeem**.
Funding acquisition: Yuegao Hu.
Investigation: Tariful Alam Khan.
Project administration: Yuegao Hu.
Resources: Lili Chen, Xiaofen Wang, Yuegao Hu.
Supervision: Xiaofen Wang ... , Yuegao Hu.
Validation: Xiaofen Wang.
Visualization: Zhaohai Zeng, Yuegao Hu.
Writing ? original draft: Tariful Alam Khan.
Writing ? review & editing: **Faisal** **Nadeem**, Yuegao Hu.
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A graph $G$ is called edge-magic if there exists a bijective function $\phi:V(G)\cup E(G)\to\{1, 2,\dots,|V(G)|+|E(G)|\}$ such that $\phi(x)+\phi(xy)+\phi(y)=c(\phi)$ is a constant for every edge $xy\in E(G)$, called the valence of $\phi$. Moreover, $G$ is said to be super edge-magic if $\phi(V(G))=\{1,2,\dots,|V(G)|\}.$ The super edge-magic deficiency of a graph $G$, denoted by...

In this paper we consider graphs in which any pair of vertices is missed by some longest path. We are proving the existence of such graphs in the infinite triangular, square and hexagonal lattices in the plane. Moreover, we extend our investigation to lattices on several surfaces such as the torus, the M\"obius strip and the Klein bottle.