Steiner trees and higher geodecity

Let $G$ be a connected graph and $\ell : E(G) \to \mathbb{R}^+$ a length-function on the edges of $G$. The Steiner distance $\mathrm{sd}_G(A)$ of $A \subseteq V(G)$ within $G$ is the minimum length of a connected subgraph of $G$ containing $A$, where the length of a subgraph is the sum of the lengths of its edges. It is clear that every subgraph $H \subseteq G$, with the induced length-function $\ell|_{E(H)}$, satisfies $\mathrm{sd}_H(A) \geq \mathrm{sd}_G(A)$ for every $A \subseteq V(H)$. We call $H \subseteq G$ $k$-geodesic in $G$ if equality is attained for every $A \subseteq V(H)$ with $|A| \leq k$. A subgraph is fully geodesic if it is $k$-geodesic for every $k \in \mathbb{N}$. It is easy to construct examples of graphs $H \subseteq G$ such that $H$ is $k$-geodesic, but not $(k+1)$-geodesic, so this defines a strict hierarchy of properties. We are interested in situations in which this hierarchy collapses in the sense that if $H \subseteq G$ is $k$-geodesic, then $H$ is already fully geodesic in $G$. Our first result of this kind asserts that if $T$ is a tree and $T \subseteq G$ is 2-geodesic with respect to some length-function $\ell$, then it is fully geodesic. This fails for graphs containing a cycle. We also prove that if $C$ is a cycle and $C \subseteq G$ is 6-geodesic, then $C$ is fully geodesic. We present an example showing that the number six is indeed optimal. We then develop a structural approach towards a more general theory and present several open questions concerning the big picture underlying this phenomenon.