Slimness of graphs

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph G = (V,E), a geodesic triangle △(x, y, z) with x, y, z ∈ V is the union P (x, y)∪P (x, z)∪P (y, z) of three shortest paths connecting these vertices. A geodesic triangle △(x, y, z) is called δ-slim if for any vertex u ∈ V on any side P (x, y) the distance from u to P (x, z)∪P (y, z) is at most δ, i.e. each path is contained in the union of the δ-neighborhoods of two others. A graph G is called δ-slim, if all geodesic triangles in G are δ-slim. The smallest value δ for which G is δ-slim is called the slimness of G. In this paper, using the layering partition technique, we obtain sharp bounds on slimness of such families of graphs as (1) graphs with cluster-diameter ∆(G) of a layering partition of G, (2) graphs with tree-length λ, (3) graphs with tree-breadth ρ, (4) k-chordal graphs, AT-free graphs and HHD-free graphs. Additionally, we show that the slimness of every 4-chordal graph is at most 2 and characterize those 4-chordal graphs for which the slimness of every of its induced subgraph is at most 1.