Steiner trees and higher geodecity

Let $G$ be a connected graph and $\mathrm{\ell }:E(G)\to {\mathbb{R}}^{+}$ a length-function on the edges of $G$. The Steiner distance ${\mathrm{s}\mathrm{d}}_G(A)$ 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 $\mathrm{\ell }{\mathrm{\mid }}_{E(H)}$, satisfies $$\mathrm{s}\mathrm{d}$_H(A) \geq $\mathrm{s}\mathrm{d}$_G(A)$forevery$A \subseteq V(H)$\mathrm{.}Wecall$H \subseteq G$ $k$-geodesic in $G$ if equality is attained for every $A\subseteq V(H)$ with $\mathrm{\mid }A\mathrm{\mid }\le k$. A subgraph is fully geodesic if it is $k$-geodesic for every $k \in $\mathbb{N}$$\mathrm{.}Itiseasytoconstructexamplesofgraphs$H \subseteq G$suchthat$H$is$k$-geodesic,butnot$(k+1)$-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 $T \subseteq G$is2-geodesicwithrespecttosomelength-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.