Map graphs having witnesses of large girth

A half-square of a bipartite graph $B=(X,Y,E_B)$ has one color class of $B$ as vertex set, say $X$; two vertices are adjacent whenever they have a common neighbor in $Y$. If $G=(V,E_G)$ is the half-square of a planar bipartite graph $B=(V,W,E_B)$, then $G$ is called a map graph, and $B$ is a witness of $G$. Map graphs generalize planar graphs, and have been introduced and investigated by Chen, Grigni and Papadimitriou [STOC 1998, J. ACM 2002]. They proved that recognizing map graphs is in $\mathsf{NP}$ by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizing map graphs is in $\mathsf{P}$, by providing an $\Omega(n^{120})$-time algorithm for $n$-vertex input graphs. In this note, we give good characterizations and efficient recognition for half-squares of bipartite graphs with girth at least a given integer $g\ge 8$. It turns out that map graphs having witnesses of girth at least $g$ are precisely the graphs whose vertex-clique incidence bipartite graph is planar and of girth at least $g$. Our structural characterization implies an $O(n^2m)$-time algorithm for recognizing if a given $n$-vertex $m$-edge graph $G$ is such a map graph.