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I am working on the following homework exercise:

Let $G = (V,E)$ be an undirected graph and $c: E \rightarrow \mathbb{R}$ it's cost function. Further let $T = (V,E')$ be a spanning tree in G.

I need to show the equivalence of the following statements:

  1. $T$ is a MST.
  2. For all $e = \{x,y\} \in E \setminus E'$ holds: No edge of the (unique) path from $x$ to $y$ has a higher cost than $e$.
  3. For all $e \in E'$ holds: Let $C$ one of the two connected components of $T-e$, then $e$ is an edge with minimal weight in $\delta(V(C))$.

I have done $1) \implies 2)$ and $2) \implies 3)$, but I do not know how to prove $3) \implies 1)$. Could you give me a hint?

EDIT: Please note that $\delta$ refers to the cut, i.e.:

$$\delta(X) = \{ \ \{u,v\} \in E(G) : u \in X \text{ and } v \notin X \}\}$$

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  • $\begingroup$ What is $\delta(V(C))$? I cannot find the definition of $\delta$ anywhere. $\endgroup$ – Apass.Jack Oct 15 '18 at 6:56
  • $\begingroup$ It is the cut: en.wikipedia.org/wiki/Cut_(graph_theory) $\endgroup$ – 3nondatur Oct 15 '18 at 6:58
  • $\begingroup$ That is what I suspected. However, as in the Wikipedia entry, a cut or a cut-set is usually specified by both vertex subsets of a partition of $V$. Do you mean the cut-set $(V(C), V\setminus V(C)$? $\endgroup$ – Apass.Jack Oct 15 '18 at 7:04
  • $\begingroup$ Yes, I will edit my question accordingly; sorry for the inconvenience. $\endgroup$ – 3nondatur Oct 15 '18 at 7:11
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In order to make sure $3)\implies1)\,$ is true, let us assume that all costs of edges are different. Otherwise, there is a counterexample at Is a set of acyclic |V| - 1 light edges always a Minimum Spanning Tree?.

Here is a hint to prove $3) \implies 1)\,$. Suppose you have some MST $T'$. Show that $T$ and $T'$ have the same number of edges. Show that $T'$ must contain every minimal cut.

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