On the hyperbolicity of bipartite graphs and intersection graphs

Hyperbolicity is a measure of the tree-likeness of a graph from a metric perspective. Recently, it has been used to classify complex networks depending on their underlying geometry. Motivated by a better understanding of the structure of graphs with bounded hyperbolicity, we here investigate on the hyperbolicity of bipartite graphs. More precisely, given a bipartite graph B=(V0∪V1,E) we prove it is enough to consider any one side Vi of the bipartition of B to obtain a close approximate of its hyperbolicity δ(B) — up to an additive constant 2. We obtain from this result the sharp bounds δ(G)−1≤δ(L(G))≤δ(G)+1 and δ(G)−1≤δ(K(G))≤δ(G)+1 for every graph G, with L(G) and K(G) being respectively the line graph and the clique graph of G. Finally, promising extensions of our techniques to a broader class of intersection graphs are discussed and illustrated with the case of the biclique graph BK(G), for which we prove (δ(G)−3)/2≤δ(BK(G))≤(δ(G)+3)/2.