Map graphs having witnesses of large girth
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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 NP by proving the existence of a witness. Soon later, Thorup [FOCS 1998] claimed that recognizing map graphs is in P, by providing an Ω(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> 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.
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