The main subject of metric graph theory is the investigation and
structural characterization of graph classes whose metric
satisfies the main metric and convexity properties of classical
metric geometries. The main classes of graphs occurring in metric
graph theory are median graphs, Helly graphs, bridged graphs,
hyperbolic graphs, weakly modular graphs, and isometric subgraphs
of hypercubes. Other classes of graphs in MGT occur from
combinatorics and geometry (basis graphs of matroids and dual
polar graphs) and have been characterized using metric
conditions.
Many of these classes of graphs give rise to important cubical
and simplicial complexes. Median graphs are exactly the
1--skeletons of CAT(0) cubical complexes; CAT(0) cubical
complexes were characterized by M. Gromov in a local-to-global
combinatorial way as the simply connected cubical complexes in
which the links of vertices are simplicial flag
complexes. Analogously to median graphs, bridged graphs are
exactly the 1--skeletons of simply connected simplicial flag
complexes in which the links of vertices do not contain induced
4-- and 5--cycles. Those simplicial complexes were rediscovered
by T. Januszkiewicz and J. Swiatkowski and dubbed systolic
complexes. Systolic complexes satisfy many global properties of
CAT(0) spaces (contractibility, fixed point property) and were
suggested as a variant of simplicial complexes of combinatorial
nonpositive curvature. More recently, similar local-to-global
characterizations have been obtained for basis graphs of
matroids, Helly graphs, weakly modular graphs and their
subclasses.
In the talk, we will overview some of those results and describe
two methods for proving them: minimal disk diagrams and universal
covers. The talk is based on the following papers:
J. Chalopin, V. Chepoi, H. Hirai, and
D. Osajda,
Weakly
modular graphs and nonpositive curvature (submitted).
J. Chalopin, V. Chepoi, and
D. Osajda,
On two
conjectures of Maurer concerning basis graphs of matroids,
J. Comb. Theory, Ser. B 114 (2015), 1-32.
V. Chepoi,
Graphs of some CAT(0) complexes, Adv. Appl. Math. 24 (2000),
125-179.