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Showing papers on "Adjacency list published in 1976"


Journal ArticleDOI
TL;DR: In this article, a category-theoretical approach to graphs is used to define and study double cover projections, and an upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2.
Abstract: A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

47 citations


Journal ArticleDOI
TL;DR: In this article, a procedure is outlined which allows the symmetry properties of graphs to be systematically and rigorously investigated, based on a search for all the automorphisms of a graph and this is accompanied by suitably applying the procedure for recognizing identical graphs.

42 citations


Journal ArticleDOI
TL;DR: Motion induction was investigated as a function of depth adjacency and attention and it was found that separating the test and induction object in apparent depth decreased the induction effect.

41 citations


Journal ArticleDOI
TL;DR: In this paper, a necessary and sufficient condition for adjacency on the convex hull of 0-1 feasible points was given for the set partitioning problem, and a strong bound was derived for the diameter of the polytope associated with the conveX hull of $0 -1$ feasible points.
Abstract: A necessary and sufficient condition is given for adjacency on the convex hull of 0–1 feasible points. The class of problems for which this condition is valid includes the set partitioning problem. A strong bound is derived for the diameter of the polytope associated with the convex hull of $0 - 1$ feasible points. A counterexample is given to a published necessary and sufficient condition for nonadjacency of two traveling salesman tours on their convex hull. A necessary condition is obtained for two tours to be nonadjacent on their convex hull. A sufficient condition for nonadjacency is also given. Examples are provided for the traveling salesman problem to show that neither the necessary condition is sufficient nor the sufficient condition is necessary. Finally, some adjacency properties are given for the traveling salesman tours on the assignment polytope.

27 citations


Journal ArticleDOI
TL;DR: A corresponding algorithm based on the removal of a line from a graph is developed and simple proofs of algorithms for a list to be multigraphical due to Hakimi and Butler are provided.
Abstract: An important and basic characterization of a graph is the sequence or list of degrees of that graph. Problems regarding the construction of graphs with specified degrees occur in chemistry and in the design of reliable networks. A list of nonnegative integers is called graphical if there is a graph (called a realization) with the given list as its degree list. The usual algorithms for determining whether a given list is graphical are derived from the effect on a graphical list of the removal of a point from a graph. After reviewing such an algorithm by Havel-Hakimi and its generalization by Wang and Kleitman, we develop a corresponding algorithm based on the removal of a line from a graph. We conclude by reviewing and providing simple proofs of algorithms for a list to be multigraphical due to Hakimi and Butler. The conditions relating a graphical or multigraphical list to the point and line connectivity of their realizations, due to Edmonds, Wang and Kleitman, Boesch and McHugh, and Hakimi, are presented along with new and simple proofs of the multigraph case.

18 citations


Proceedings ArticleDOI
07 Jun 1976
TL;DR: Bit map and list structures are analyzed for representation of a network, using its adjacency matrix, and conversion algorithms are presented, designed to minimize extra work space.
Abstract: Bit map and list structures are analyzed for representation of a network, using its adjacency matrix. Storage analysis, with reduction for m-partite networks, reveals a fundamental function of problem dimensions, called the "index threshold." Several examples, from industry and government, are cited to illustrate the analysis. Relative processing merits are studied, and an equation is derived to relate "processing ratio" to the ratio of the index threshold to the index size. Finally, conversion algorithms are presented, designed to minimize extra work space.

3 citations



01 May 1976
TL;DR: In this article, the simplex method is viewed as a process of following a trajectory and a series of definitions, lemmas and theorems are given to make precise such notions as basic solution, distinct solution, adjacency, and dual basis.
Abstract: In Part I, the classical statement of an LP problem is compared with the most general form which general-purpose LP software can usually accept. The latter form is then simplified to the form used internally by such software. An extended matrix representation of the conditions used in the simplex method is given, plus a list of the various outcomes of pivot selection. All this is merely a review and summary in consistent notation. The remainder of Part I views an LP problem as a function of its objective form and parametric algorithms as families of functions. The simplex method, as a process, is also viewed as following a trajectory. The ambiguity of extending this idea to the dual feasible subspace is indicated as well as the difficulty of using this viewpoint for integer programs. Part II begins with a fairly complete list of notation required in discussing details of the simplex method and its variants. Then a series of definitions, lemmas and theorems are given to make precise such notions as basic solution, distinct solution, adjacency, and dual basis. The main result is a clarification of the phenomena of degeneracy and alternate solutions, in both primal and dual senses. In particular, the complementary nature of ambiguous solutions and multiple solutions is shown. Two trivial examples, easily followed, are sufficient to illustrate these ideas. Part III applies the ideas of Part II, plus one other, to the old problems of exploring the vicinity of optimality, resolving revised models from an old basis, and a few special problems for which the simplex method is sometimes useful in a non-LP context.

1 citations