id: 06026831
dt: j
an: 06026831
au: Bermudo, Sergio; Rodríguez, José M.; Sigarreta, José M.
ti: Computing the hyperbolicity constant.
so: Comput. Math. Appl. 62, No. 12, 4592-4595 (2011).
py: 2011
pu: Elsevier Science Ltd. (Pergamon), Oxford
la: EN
cc: 
ut: graph; geodesic metric space; triangle; hyperbolic space; hyperbolicity
    constant
ci: 
li: doi:10.1016/j.camwa.2011.10.041
ab: If $X$ is Gromov’s $δ$-hyperbolic space, then the hyperbolicity constant
    of $X$ is $\inf\{δ\geq 0: X\text{ is }δ\text{-hyperbolic}\}$. The
    authors study graphs with edges of the same length $k$. If $G$ is such
    graph, then $J(G)$ denotes the union of the set $V(G)$ of vertices of
    $G$ and the midpoints of the edges of $G$. Let $T_1$ denote the set of
    all geodesic triangles with vertices in $J(G)$ and which are simple
    cycles (i.e., simple closed curves). Then $δ_1(T)=\inf \{λ\geq
    0:\text{ every triangle }T\in T_1\text{ is }λ\text{-thin}\}$. The main
    result of the paper states that for every graph $G$ with edges of equal
    length $k$, $δ_1(G)=δ(G)$. In addition, it is shown that the
    hyperbolicity constant $δ(G)$ is a multiple of $k/4$. As a corollary,
    the authors derive that for every hyperbolic graph $G$ with edges of
    length $k$, there is a triangle $T\in T_1$ for which the hyperbolicity
    constant is attained.
rv: I. G. Nikolaev (Urbana)