Slimness of graphs

Abstract

Slimness of a graph measures the local deviation of its metric from a tree metric. In a graph $G=(V,E)$, a geodesic triangle $\bigtriangleup(x,y,z)$ with $x, y, z\in V$ is the union $P(x,y) \cup P(x,z) \cup P(y,z)$ of three shortest paths connecting these vertices. A geodesic triangle $\bigtriangleup(x,y,z)$ is called $δ$-slim if for any vertex $u\in V$ on any side $P(x,y)$ the distance from $u$ to $P(x,z) \cup P(y,z)$ 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 $Δ(G)$ 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.