0.3 The steiner tree problem  (Page 2/3)

Proof

The Euler characteristic for lines is 1, which means that $\#\text{vertices}-\#\text{edges}=1$ . Therefore there are $n+k-1$ edges in a tree, where $n$ is the number of terminal points and $k$ is the number of Steiner points.

Each Steiner point has 3 edges coming from it, and each terminal point has at least one, so there are at least $\frac{3k+n}{2}$ total edges; the division by 2 accounts for the fact that each edge is counted at two vertices. Therefore

which, by simple algebra, yields

Definition

A Steiner tree with $n-2$ Steiner points is a full Steiner tree.

Theorem 5

Every Steiner tree that is not a full Steiner tree can be decomposed into a union of full Steiner trees.

Proof

See [link]

Examples

In a 3 by 4 rectangle, two Steiner trees can be created, but the one with edge between the Steiner points parallel to the long side is the Steiner minimal tree.

In a 1 by 2 rectangle, the minimum spanning tree is not a Steiner tree.

Definition

A topology on a graph determines how many terminals and Steiner points there are, and how the points are connected to each other. A topology is a degeneracy of another if it can be obtained from shrinking edges of the original. The set of degeneracies of a topology $G$ is denoted by $D\left(G\right)$ .

A topology is a Steiner topology (full Steiner topology) if every Steiner point has degree 3 (and terminal has degree 1).

If $F$ is a full Steiner topology, then $D_s\left(F\right)$ is the set of Steiner topologies in $D\left(F\right)$ .

It is easy to see that a Steiner tree contains no closed shapes, called cycles. Otherwise, one edge could be deleted, and the resulting structure would be of shorter total length while still connecting all of the points.

Theorem 5

A Steiner tree whose topology is in $D\left(F\right)$ for some full Steiner topology $F$ is the unique minimum tree among trees whose topologies are in $D_s\left(F\right)$ .

Proof

Let $T$ be a tree whose topology $G\in D_s\left(F\right)$ . Let $v_1,...,v_n$ be an enumeration of the vertices. Define $f_{ij}=1$ if $[v_i,v_j]$ is an edge in $G$ and $f_{ij}=0$ otherwise. Then the length of $T$ is

$|v_i-v_j|$ is a norm, and norms are convex due to the triangle inequality, so $T$ is strictly convex except when edges remain the same direction in a perturbation. In that case, when the three edges from a Steiner point are parallel, moving the Steiner point towards the side of the two overlapping edges decreases the total length, and thus $\left|T\right|$ is strictly convex. Because a Steiner tree is a local minimum, strict convexity guarantees that it is the unique minimum among trees with topologies in $D_s\left(F\right)$ .

Corollary

There exists at most one relatively minimal tree for a given topology.

Definition

A Steiner hull for a given set of points $N$ is a region which is known to contain a Steiner minimal tree.

In the Euclidean plane, the Steiner hull is bounded.

Lemma

Let $[v_i,v_j]$ be any path in a Steiner minimal tree, where $v_i$ and $v_j$ are defined as before. Let $L(v_i,v_j)$ be the region consisting of all points $p$ satisfying

$L(v_i,v_j)$ is the lune-shaped intersection of circles of radius $|v_i-v_j|$ centered on $v_i$ and $v_j$ . No other vertex of the Steiner minimal tree can lie in $L(v_i,v_j)$ .