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I'm trying to understand a proof from the book "Graph Theory with Application to Engineering & Computer Science" by Narsingh Deo.

The chapter is about trees in non oriented graphs.

A bit of terminology so that you can understand the theorem and the beginning of the proof from the book:

The author calls minimum spanning trees, shortest spanning trees.

The author calls a branch of a spanning tree any edge of the tree.

A chord of a spanning tree is any edge of the underlying graph that is not in the tree.

A fundamental circuit associated to a spanning tree is a circuit formed by adding one of its chords to a spanning tree (for the author a "circuit" is a closed path, there is no repetition of vertices, it's what most other sources I've read call a cycle). So, a fundamental circuit associated to a spanning tree is a actually a cycle formed by adding one of its chords to a spanning tree.

The distance between two spanning trees $S$ and $T$ of the same graph is (regarding $S$ and $T$ as sets of edges), is $|S\setminus T|$ (which happens to be equal to $|T\setminus S|$).

There's a step in the proof of Theorem 3-16 that I don't understand.

Theorem 3-16:

A spanning tree T (of a given weighted connected Graph G) is a shortest spanning tree (of G) if and only if there exists no other spanning tree (of G) at a distance of one from T whose weight is smaller than that of T

Proof:

Let $T_1$ be a spanning tree in G satisfying the hypothesis of the theorem (i.e. there is no spanning tree at a distance of one from $T_1$ which is shorter than $T_1$). The proof will be completed by showing that if $T_2$ is a shortest spanning tree (different from $T_1$) in G, the weight of $T_1$ will also be equal to that of $T_2$. Let $T_2$ be a shortest spanning tree in G. Clearly, $T_2$ must also satisfy the hypothesis of the theorem (otherwise there will be a spanning tree shorter than $T_2$ at a distance of one from $T_2$, violating the assumption that $T_2$ is shortest).

Consider an edge $e$ in $T_2$ which is not in $T_1$. Adding $e$ to $T_1$ forms a fundamental circuit with branches in $T_1$. Some, but not all, of the branches in $T_1$ that form the fundamental circuit with $e$ may also be in $T_2$; each of these branches in $T_1$ has a weight smaller than or equal to that of $e$, because of the assumption on $T_1$. Amongst all those edges in this circuit which are not in $T_2$ at least one, say $b_j$, must form some fundamental circuit (with respect to $T_2$) containing $e$.

I'm stuck at the last sentence I just quoted:

"Amongst all those edges in this circuit which are not in $T_2$ at least one, say $b_j$, must form some fundamental circuit (with respect to $T_2$) containing $e$."

I don't see why among those cycles, there should necessarily be one that contains $e$. Why is that?

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If we remove $e = (x,y)$ from $T_2$, then it breaks into two connected components, say $T_{2,x},T_{2,y}$. There is a unique path in $T_1$ from $x$ to $y$, which together with $e$ forms a fundamental cycle of $T_1$. The path starts at $T_{2,x}$ and ends at $T_{2,y}$, hence at some point it crosses between the two. Adding the corresponding edge to $T_2$ would result in a fundamental cycle containing $e$.

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  • $\begingroup$ I don't get it. The path you're talking about is a path in $T_1$. Why would there be an edge in that path, that, when added to $T_2$ would form, along with some other edges of $T_2$, a cycle containing $e$? $\endgroup$ – DisplayName80 Oct 21 '20 at 15:02
  • $\begingroup$ If I follow you, there's that edge, let's call it $f$, in $T_1$, that reconnects the two components of $T_2 \setminus \{e\}$. What proves that the fundamental cycle in $T_2$ associated to $f$ contains $e$ ? $\endgroup$ – DisplayName80 Oct 21 '20 at 15:48
  • $\begingroup$ I think I'm starting to understand. I'm going to try to detail the properties you're using. $\endgroup$ – DisplayName80 Oct 21 '20 at 16:09
  • $\begingroup$ If I remove $e = \{x, y\}$ from $T_2$, I get two connected components $T_{2, x}$, $T_{2, y}$ with $x \in T_{2, x}$ and $y \in T_{2, y}$. If I remove $e$ from the fundamental cycle it forms with edges of $T_1$, I get a path $P_1$ from $x$ to $y$ composed of edges of $T_1$. One of these edges must have one end in each component $T_{2, x}$, $T_{2, y}$ (because any path that starts in a set of vertices and ends in another set of vertices must contain an edge with one extremity in each set). I call that edge $f = \{x', y'\}$, with $x' \in T_{2, x}$ and $y' \in T_{2, y}$. $\endgroup$ – DisplayName80 Oct 21 '20 at 19:26
  • $\begingroup$ $f$ does not belong to $T_2$, otherwise, since $f$ connects $T_{2, x}$ and $T_{2, y}$, and since $f \neq e$, $T_2 \setminus \{e\}$ would be connected. $\endgroup$ – DisplayName80 Oct 21 '20 at 19:26

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