Bipartite half

Summary

In graph theory, the bipartite half or half-square of a bipartite graph $G = (U,V,E)$ is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, $U$) and in which there is an edge $u_{i}u_{j}$ for each pair of vertices $u_{i}, u_{j}$ in $U$ that are at distance two from each other in $G$. That is, in a more compact notation, the bipartite half is $G^{2}[U]$ where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph.