Abstract. We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. In particular, given a mapping, we show how to embed two paths on an n × n grid, and two caterpillar graphs on a 3n × 3n grid. We show that it is not always possible to simultaneously embed three paths. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2 ) × O(n 2 ) grid.

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n × n grid, and two caterpillar graphs on a 3n × 3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2 ) × O(n 2 ) grid.

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Graph Theory is still a relatively young subject, and debate still rages on what material constitutes the core results that any introductory text should include. Bollobás has chosen to introduce graph theory - including recent results - in a way that emphasizes the connections between (for example) the Tutte polynomial of a graph, the partition functions of theoretical physics, and the new knot polynomials, all of which are interconnected.On the other hand, graph theory is also rooted strongly in computing science, where it is applied to many different problems; Bollobás's treatment is completely theoretical and does not address these applications. Or, in more practical terms, he is concerned whether a solution exists, rather than asking whether the solution can be computed in a reasonably efficient manner.One of the pleasures of working in graph theory is the abundance of problems available to solve. Unlike many more traditional areas of Mathematics, knowing the core results and proofs is frequently insufficient. Often solving a new problem requires a new approach, or a subtle twist on an existing one, combined with some bare knuckle work. Bollobás emphasizes this in the problems available at the end of each chapter; he includes in total 639 problems, ranging from the reasonably straightforward to the very difficult. I spent time with friends working on these problems, and was intrigued by the variety of the proofs that we came up with.

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The study of computability and complexity theory lies at the root of computing science, as computer scientists try to address the dual questions of which problems are in fact solvable, and how much effort should we expect to expend to solve a given problem.Problems are grouped together in classes on the basis of these questions. Thus P is the class of problems that can be solved in polynomial time, while NP is the class of problems that can be accepted in polynomial time using a non-deterministic Taring Machine. NP-complete problems were first studied in the sixties and early seventies.