Algorithmic graph theory and perfect graphs

Several data presentation problems involve drawing graphs so that they are easy to read and understand. Examples include circuit schematics and diagrams for information systems analysis and design. In this paper we present a bibliographic survey on algorithms whose goal is to produce aesthetically pleasing drawings of graphs. Research on this topic is spread over the broad spectrum of computer science. This bibliography constitutes a first attempt to encompass both theoretical and application-oriented papers from disparate areas.

Algorithms for drawing graphs: an annotated bibliography

Vertex-to-vertex pursuit in a graph

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Shellable and Cohen-Macaulay partially ordered sets

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

Art Gallery and Illumination Problems

I Graphical Data Structures.- Graphical data structures for ordered sets.- Lattices in data analysis: how to draw them with a computer.- A computer program for orthomodular lattices.- II Computational Complexity.- Computationally tractable classes of ordered sets.- The complexity of orders.- The calculation of invariants for ordered sets.- III Sorting and Scheduling.- Data manipulations based on orderings.- Preemptive scheduling.- IV Enumeration.- Enumeration of ordered sets.- 0-1 laws in logic and combinatorics.- V Geometry.- Partial orders and Euclidean geometry.- VI Decision Making.- Human decision making and ordered sets.- VII Problem Sessions.- ORDER's problem list.- Scheduling.- The diagram.- Linear extensions.- Enumeration.- Sorting.- Miscellany.- VIII IndeX.

Algorithms and Order

Graphs and Order

Let G be an undirected graph without multiple edges and with a loop at every vertex—the set of edges of G corresponds to a reflexive and symmetric binary relation on its set of vertices. Then every edge-preserving map of the set of vertices of G to itself fixes an edge [{f(a), f(b)} = {a, b} for some edge (a, b) of G] if and only if (i) G is connected, (ii) G contains no cycles, and (iii) G contains no infinte paths. The proof is conerned with those subgraphs H of a graph G for which there is an edge-preserving map f of the set of vertices of G onto the set of vertices of H and satisfying f(a) = a for each vertex a of H.

Fixed-edge theorem for graphs with loops

A fixed point theorem for finite partially ordered sets

For a finite ordered set P how can a linear extension L = Cl D C2 D... Cm,, be constructed which minimizes the number m of chains C, of P? While this question remains largely unanswered we show that a natural "greedy" algorithm is actually optimal for a far wider class of ordered sets than was hitherto suspected. The most fundamental and yet far-reaching results in the theory of ordered sets are the following: (R. P. Dilworth [1950]) In a finite ordered set P, the minimum number of disjoint chains whose set union is all of P equals the maximum number of pairwise noncomparable elemnents of P. (E. Szpilrajn [1930]) Every ordered set has a linear extension. Let C1, C2,. . . ,CCm be any minimum family of disjoint chains of a finite ordered set P whose set union is P. The linear sum C1 e C2 e ... e Cm of these chains need not be a linear extension of P. On the other hand, any linear extension L of a finite ordered set P can be expressed as the linear sum C1 e C2 3 * * Cm of chains Ci in P (so chosen that supp Cl 4 infp Cl+ 1 for each i) and, while Uk I Ci = P, this family of chains need not be the minimum number whose set union is P (see Figure 1). How can these two fundamental results about ordered sets be reconciled? Indeed, until recently there seemed little reason even to consider such a question.

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Ivan Rival

Papers

Algorithms and Order

Graphs and Order

Fixed-edge theorem for graphs with loops

A fixed point theorem for finite partially ordered sets

Optimal linear extensions by interchanging chains