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If X is a geodesic metric space and $$x_1,x_2,x_3$$ are three points in $$X$$ , a geodesic triangle $$T=\{x_1,x_2,x_3\}$$ is the union of three geodesics $$[x_1x_2]$$ , $$[x_2x_3]$$ and $$[x_3x_1]$$ in $$X$$ . The space $$X$$ is $$\delta $$ - hyperbolic $$($$ in the Gromov sense $$)$$ if any side of $$T$$ is contained in a $$\delta $$ -neighborhood of the union of the two other sides, for every geodesic triangle $$T$$ in $$X$$ . The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. In this paper we obtain criteria which allow us to decide, for a large class of graphs, whether they are hyperbolic or not. We are especially interested in the planar graphs which are the 'boundary' (the $$1$$ -skeleton) of a tessellation of the Euclidean plane. Furthermore, we prove that a graph obtained as the $$1$$ -skeleton of a general CW $$2$$ -complex is hyperbolic if and only if its dual graph is hyperbolic. [ABSTRACT FROM AUTHOR]

If X is a geodesic metric space and x1,x2,x3∈X, a geodesic triangleT={x1,x2,x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e. δ(X)=inf{δ≥0:Xisδ-hyperbolic}. In this paper, we characterize the lexicographic product of two graphs G1∘G2 which are hyperbolic, in terms of G1 and G2: the lexicographic product graph G1∘G2 is hyperbolic if and only if G1 is hyperbolic, unless if G1 is a trivial graph (the graph with a single vertex); if G1 is trivial, then G1∘G2 is hyperbolic if and only if G2 is hyperbolic. In particular, we obtain the sharp inequalities δ(G1)≤δ(G1∘G2)≤δ(G1)+3/2 if G1 is not a trivial graph, and we characterize the graphs for which the second inequality is attained. [ABSTRACT FROM AUTHOR]

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G 1 × G 2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs). [ABSTRACT FROM AUTHOR]