Graphs of some CAT(0) complexes

In this note, we characterize the graphs (1-skeletons) of some piecewise Eu clidean simplicial and cubical complexes having nonpositive curvature in th e sense of Gromov's CAT(0) inequality. Each such cell complex K is simply c onnected and obeys a certain flag condition. It turns out that if, in addit ion, all maximal cells are either regular Euclidean cubes or right Euclidea n triangles glued in a special way, then the underlying graph G(K) is eithe r a median graph or a hereditary modular graph without two forbidden induce d subgraphs. We also characterize the simplicial complexes arising from bri dged graphs, a dass of graphs whose metric enjoys one of the basic properti es of CAT(0) spaces. Additionally, we show that the graphs of all these com plexes and some more general classes of graphs have geodesic combings and b icombings verifying the 1- or 2-fellow traveler property. (C) 2000 Academic Press.