Bipartite half



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 $uiuj$ for each pair of vertices $ui, uj$ in $U$ that are at distance two from each other in $G$. That is, in a more compact notation, the bipartite half is $G2[U]$ where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph.

Examples
For instance, the bipartite half of the complete bipartite graph $Kn,n$ is the complete graph $Kn$ and the bipartite half of the hypercube graph is the halved cube graph. When $G$ is a distance-regular graph, its two bipartite halves are both distance-regular. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs.

Representation and hardness
Every graph $G$ is the bipartite half of another graph, formed by subdividing the edges of $G$ into two-edge paths. More generally, a representation of $G$ as a bipartite half can be found by taking any clique edge cover of $G$ and replacing each clique by a star. Every representation arises in this way. Since finding the smallest clique edge cover is NP-hard, so is finding the graph with the fewest vertices for which $G$ is the bipartite half.

Special cases
The map graphs, that is, the intersection graphs of interior-disjoint simply-connected regions in the plane, are exactly the bipartite halves of bipartite planar graphs.