TAOCP 7.2.2.1 Exercise 325

Section 7.2.2.1: Dancing Links

Exercise 325. [27] [27] The Somap is the graph whose vertices are the 240 distinct solutions to the Soma cube problem, with $u — v$ if and only if $u$ can be obtained from an equivalent of $v$ by changing the positions of at most three pieces. The irtex Somap is similar, but it has $u — v$ only when a change of just two pieces gets from one to the other.

a) What are the degree sequences of the Somap graphs? b) How many connected components do they have? How many bicomponents?

Verified: no
Solve time: 2m20s

Setup

Let $V$ be the set of $240$ equivalence classes of solutions of the Soma cube problem. The two graphs in the exercise are defined on the same vertex set $V$.

For $u,v\in V$, the Somap relation requires checking whether some representative of $u$ can be obtained from some representative of $v$ by changing the positions of at most three pieces. The strong Somap relation uses the same condition with at most two pieces.

To give the requested degree sequences and component data, one must have an explicit enumeration of the $240$ solutions together with a canonical representation of each solution and a procedure for testing whether two solutions differ by moving a specified number of pieces.

Solution

The information supplied in the question and the quoted section is insufficient to derive the numerical answers. Section 7.2.2.1 defines the Somap graphs, but it does not provide the $240$ solution representatives, the adjacency matrix of either graph, or any intermediate enumeration data from which the degrees and bicomponents can be computed.

A rigorous solution requires the following finite computation.

For each pair $(u,v)$ of the $240$ solutions, enumerate all choices of up to three pieces whose positions are changed, normalize the resulting configuration by the allowed cube symmetries, and test whether the resulting solution is equivalent to $u$. This gives the adjacency matrix of the Somap. Repeating the same computation with at most two changed pieces gives the adjacency matrix of the strong Somap.

For each obtained adjacency matrix, the degree of a vertex $v$ is the number of vertices adjacent to $v$. Sorting these $240$ degrees gives the required degree sequence.

The connected components are obtained by a graph traversal. The bicomponents are obtained by applying the standard depth-first search algorithm for articulation points and biconnected components to the resulting graph.

The numerical degree sequences and component counts therefore depend on the explicit list of the $240$ Soma cube solutions and cannot be derived from the Section 7.2.2.1 excerpt alone.

Verification

The missing data are essential, not cosmetic. Two different enumerations of $240$ objects with the same definition of adjacency can only be distinguished by the actual solution configurations. The degree sequence is a property of the edge set, and the edge set is not determined by the definition without carrying out the finite enumeration.

A complete answer must therefore include either the generated list of all $240$ solutions or the resulting adjacency data before the final numerical values can be stated and boxed.

Notes

The exercise is a computational enumeration problem rather than a purely deductive consequence of the dancing-links discussion. Algorithm X and dancing links provide a method for generating the Soma cube solutions, but the graph invariants asked for in this exercise require an additional graph-analysis stage after enumeration.