Conditions on square geometric graphs

For any metric $d$ on $\mathbb{R}^2$, an ($\mathbb{R}^2,d$)-geometric graph is a graph whose vertices are points in $\mathbb{R}^2$, and two vertices are adjacent if and only if their distance is at most 1. If $d=\|.\|_{\infty}$, the metric derived from the $L_{\infty}$ norm, then $(\mathbb{R} ^2,\|.\|_{\infty})$-geometric graphs are precisely those graphs that are the intersection of two unit interval graphs. We refer to $(\mathbb{R}^2,\|.\|_{\infty})$-geometric graphs as square geometric graphs. We represent a characterization of square geometric graphs. Using this characterization we provide necessary conditions for the class of square geometric $B_{a,b}$-graphs, a generalization of cobipartite graphs. Then by applying some restrictions on these necessary conditions we obtain sufficient conditions for $B_{a,b}$-graphs to be square geometric.

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