Let G be a connected graph and $\ell :E\left(G\right)\to {\mathbb{R}}^{+}$ a length‐function on the edges of G . The Steiner distance sd_G (A ) of A  ⊆ 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  ⊆ G , endowed with the induced length‐function ℓ ∣_{E (H )}, satisfies sd_H (A ) ≥ sd_G (A ) for every A  ⊆ V (H ). Given an integer k  ≥ 2, we call H  ⊆ G k‐isometric in G if equality is attained for every A  ⊆ V (H ) with ∣A ∣ ≤ k . A subgraph is fully isometric if it is k ‐isometric for every $k\in \mathbb{N}$. It is easy to construct examples of graphs H  ⊆ G such that H is k ‐isometric, but not (k  + 1)‐isometric, so this defines a strict hierarchy of properties. We are interested in situations in which this hierarchy collapses in the sense that if H  ⊆ G is k ‐isometric, then H is already fully isometric in G . Our first result of this kind asserts that if T is a tree and T  ⊆ G is 2‐isometric with respect to some length‐function ℓ , then it is fully isometric. This fails for graphs containing a cycle. We then prove that if C is a cycle and C  ⊆ G is 6‐isometric, then C is fully isometric. We present an example showing that the number 6 is indeed optimal. We then develop a structural approach toward a more general theory and present several open questions concerning the big picture underlying this phenomenon.

中文翻译：

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Steiner距离的等距子图

令G为连通图，$\ell ：Ë（G）\to {\mathbb{[R}}^{+}$G边缘上的长度函数。在斯坦纳距离SD _ģ（甲）的甲 ⊆  V（G ^）内ģ是的连通子的最小长度ģ含甲，其中一个子图的长度是它的边缘的长度的总和。很显然，每个子图ħ  ⊆  ģ，赋有诱导长度函数ℓ | _È（ħ），满足SD _ħ（甲）≥SD _ģ（甲），用于每甲 ⊆  V（ħ）。给定的整数ķ  ≥2，我们称之为ħ  ⊆  ģK-等距G中，如果达到每平等甲 ⊆  V（ħ）与|甲|≤  ķ。如果每个子图都是k等距的，则该子图是完全等距的$ķ\in \mathbb{ñ}$。这是很容易构造图的示例ħ  ⊆  ģ使得ħ是ķ -isometric，但不是（ķ  + 1）-isometric，所以这定义特性的严格的等级。我们感兴趣的情况，即该层次结构在这个意义上，如果崩溃^ h  ⊆ 摹是ķ -isometric，然后^ h已经处于完全等距摹。我们的这种第一结果断言如果Ť是树和Ť  ⊆  ģ是2-等距相对于一些长度函数ℓ，则它是完全等距的。对于包含循环的图，此操作将失败。然后，我们证明了如果c ^是一个周期和ç  ⊆  ģ是6-等距，然后Ç完全等距。我们提供一个示例，显示数字6确实是最优的。然后，我们针对更一般的理论开发一种结构化方法，并提出了有关此现象背后的概貌的几个开放问题。