When is a graph a $\operatorname{CAT}(\kappa)$ space?

Question

Let $G:=(E,V,W)$ be a weighted graph and let $d_G$ be its graph metric, defined by on any two edges $e_1,e_2\in E$ by $$ d_G(e_1,e_2)\triangleq \inf_{\gamma}\, \sum_{v\in \gamma} W(v),\qquad\tag{0}\label{0} $$ where the infimum is taken over all sequences of vertices $\gamma=(v_1,\dots,v_t)$ connecting $e_1$ to $e_2$; where $W:V\rightarrow \mathbb{R}$ is the weight function defined on unweighted graph $(E,V)$.

In Ballmann's book Lectures on spaces of nonpositive curvature conditions are given which guarantee that $d_G$ has non-positive curvature (in the sense of Alexandrov); namely, there is some $c>0$ satisfying: $$ \inf_{v}\, W(v)\geq c.\tag{1}\label{1} $$ My question is, when is $(E,d_G)$ a $\operatorname{CAT}(\kappa)$ space, for some $\kappa\in \mathbb{R}$?

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My hypothesis is that there exist some partition $C_1,\dotsc,C_N$ of $E$ into open balls such that \eqref{1} holds; where the infimum in \eqref{0} is taken over all paths $\gamma$ contained in $C_i$; where $e_1,e_2\in C_i$.

Answer 1

The generalized Cartan--Hadamard theorem states that a length space is CAT(1) if it is locally CAT(1) + any closed curve of length $<2{\cdot}\pi$ is null-homotopic in class of curves of length $<2{\cdot}\pi$.

If the space is a graph, then the latter is equivalent to saying that any cycle has length at least $2{\cdot}\pi$.