Let B=(X,Y,E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$B=(X,Y,E)$$\end{document} be a bipartite graph. A half-square of B has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen et al., in: Proceedings of the thirtieth annual ACM symposium on the theory of computing, STOC ’98, pp 514–523. https://doi.org/10.1145/276698.276865, 1998; J ACM 49(2):127–138. https://doi.org/10.1145/506147.506148, 2002), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup’s O(n120)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$O(n^{120})$$\end{document}-time algorithm (Thorup, in: Proceedings of the 39th IEEE symposium on foundations of computer science (FOCS), pp 396–405. https://doi.org/10.1109/SFCS.1998.743490, 1998). In this paper, we identify the first hardness case, namely that deciding if a graph is a half-square of a balanced bisplit graph is NP-complete. (Balanced bisplit graphs form a proper subclass of star convex bipartite graphs). For classical subclasses of tree convex bipartite graphs such as biconvex, convex, and chordal bipartite graphs, we give good structural characterizations of their half-squares that imply efficient recognition algorithms. As a by-product, we obtain new characterizations of unit interval graphs, interval graphs, and of strongly chordal graphs in terms of half-squares of biconvex bipartite, convex bipartite, and of chordal bipartite graphs, respectively. Good characterizations of half-squares of star convex and star biconvex bipartite graphs are also given, giving linear-time recognition algorithms for these half-squares.
