Bucolic graphs and complexes
Presented by Prof. Boštjan BREšAR
Type: Oral presentation
Track: Graph Theory
Median graphs are one of the central classes of graphs in metric graph theory. They arise in different guises and applications, and relate to many other mathematical structures. Several natural non-bipartite generalizations that capture various properties of median graphs have also been studied, in particular quasi-median graphs, weakly median graphs and fiber-complemented graphs. In this talk we present recently introduced class of the (strongly) bucolic graphs, that are a common generalization of median and of bridged graphs (the latter graphs are also widely known as the graphs in which there are no isometric cycles of length more than 3). We prove that bucolic graphs are precisely the retracts of Cartesian products of weakly bridged graphs; in turn they are exactly the weakly modular graphs satisfying some local conditions formulated in terms of forbidden induced subgraphs. Moreover, finite bucolic graphs can be obtained by gated amalgamations from products of weakly bridged graphs. Bucolic complexes, whose 1-skeletons are precisely bucolic graphs, can be defined as simply connected prism complexes satisfying some local combinatorial conditions. We show that locally-finite bucolic complexes are contractible, and satisfy some nonpositive-curvature-like properties. In particular, we prove the fixed point theorem for finite group actions on such complexes. This is a joint work with J. Chalopin, V. Chepoi, T. Gologranc and D. Osajda.