Puzzles and Directed Graphs

Previously (Section 2.6) we explained how puzzle problems can be formulated in terms of graphs. The vertices of the graph correspond to the positions in a puzzle; the edges of the graph correspond to the possible moves from one position to another. The solution of the puzzle consists in finding a path from a given initial position to one (or possibly more) terminal or winning positions.

In dealing with these puzzles, we used undirected graphs. This was based upon the tacit assumption that the moves can be made both ways from one position to another. Such a procedure is permissible for the puzzles of the ferryman, the three jealous husbands, and the moves of the knight on the chessboard.

But for many puzzles the moves can only be made in one direction, and in this case we are compelled to use directed graphs in the representation. If some moves can be made in both directions, we can include an edge for each direction, or we can use a mixed graph in which these edges are undirected. To solve the puzzle, we must find a directed path from the initial position in the graph to the desired terminal position.

We shall illustrate these remarks by considering an ancient and familiar puzzle.

Relations and Graphs

So far we have discussed a variety of uses of graphs. Applications to everyday problems and to games and puzzles were considered. Our choice of topics had the advantage that we could deal with well-known and simple concepts. In this chapter, we shall strive to make clear that graphs are closely related to (indeed, are only a different way of formulating) some of the most fundamental concepts of mathematics in general.

A mathematical system, as we usually encounter it, consists of a set of objects or elements. For instance, we deal commonly with numbers and these may belong to more or less general types; we may discuss the set of integers, the positive numbers, the rational numbers, real numbers, imaginary numbers, or complex numbers. In algebra, we are concerned with elements which can be added, subtracted, multiplied, and so on. In geometry, we ordinarily have before us a set of points or special categories of points like straight lines, circles, planes, etc. In logic, we deal with the properties of statements of various kinds.

To construct a mathematical theory we need more than these elements; we need relations between them. Let us illustrate this: in the case of numbers, we have equal numbers a and b; in formal mathematical terminology, we write a = b. We also have numbers a and b which are different, and we write a ≠ b.

Conditions for Planar Graphs

As we have already explained in Section 1.4, a planar graph is a graph which can be drawn in the plane so that the edges have no intersections except at the vertices. We also gave a number of illustrations of planar graphs. In Section 1.5, we analyzed the problem of the three houses and the three wells, and explained why the corresponding graph could not be planar. The graph of the problem (Figure 1.16) can be drawn in many ways, as is possible for all graphs. When we say “the graph,” we mean any graph isomorphic to a particular graph describing the situation. Therefore, the statement “the graph in Figure 1.16 is not planar” means it has no planar isomorph. For instance, the vertices may be placed in a hexagon as in Figure 8.1. The intersection in the center is not a vertex; the edges should be considered to pass over each other at this point.

There is even a graph with only 5 vertices which is not planar—namely, the complete graph on 5 vertices (Figure 8.2). Why this graph is not planar may be made clear by reasoning similar to that used in Section 1.5 to show that the graph in Figure 8.1 (or Figure 1.16) is not planar. The vertices in any representation of the graph must lie on a cycle C in some order—say, abcdea. There is an edge eb, and in our planar graph we have the choice of putting it on the inside or the outside of C.