TAOCP 2.3.4.1 Exercise 3

The reviewer's objections are decisive.

taocpmathematicsalgorithmsvolume-1

Section 2.3.4.1: Free Trees

Exercise 3. [**] $$15$$ What walk from Start to Stop is equivalent (in the sense of Theorem K) to one traversal of the fundamental path plus one traversal of cycle $C_2$ in Fig. 32?

Verified: no
Solve time: 2m22s

The reviewer's objections are decisive. The exercise asks for the specific walk in Fig. 32, but the figure itself is not present in the material provided. Therefore no graph-specific facts can be assumed.

A correct solution must begin by reading Fig. 32 and determining:

  1. the fundamental path $P$ from Start to Stop;
  2. the actual edges and orientations of the cycle $C_2$;
  3. the common vertices where $C_2$ meets the fundamental path.

Only after those facts are established can one compute the required walk.

By Theorem K, walks are equivalent when they have the same incidence vector. Therefore the walk corresponding to

$$ P + C_2 $$

is obtained by taking the fundamental path $P$ and inserting one complete traversal of the cycle $C_2$ at any vertex of $P$ that lies on $C_2$. Since a cycle has net incidence vector zero at every vertex, traversing $C_2$ starts and ends at the same vertex, so the resulting walk still runs from Start to Stop. Its incidence vector is exactly the sum of the incidence vectors of $P$ and $C_2$.

Suppose the cycle $C_2$ meets the fundamental path at a vertex $v$. Write

$$ P=P_1P_2, $$

where $P_1$ is the part of the fundamental path from Start to $v$, and $P_2$ is the remainder from $v$ to Stop. If $Q$ denotes one traversal of $C_2$ beginning and ending at $v$, then

$$ P_1QP_2 $$

is a Start-to-Stop walk whose incidence vector is

$$ P + C_2. $$

Indeed,

$$ I(P_1QP_2) = I(P_1)+I(Q)+I(P_2) = I(P)+I(C_2), $$

where $I(\cdot)$ denotes the incidence vector.

Therefore the answer to Exercise 2.3.4.1.3 is:

$$ \boxed{\text{Take the fundamental path from Start to Stop and insert one complete traversal of }C_2\text{ at a vertex where }C_2\text{ meets the path.}} $$

To obtain the concrete edge-by-edge walk requested by the exercise, one must inspect Fig. 32 and substitute the actual edges of $C_2$ and the actual fundamental path. The information supplied here is insufficient to identify that specific walk.