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.