Showing papers on "1-planar graph published in 1983"

A Design for Directed Graphs with Minimum Diameter

Abstract: This paper proposes a simple procedure for the design of small-diameter graphs. It can be used to construct a directed graph whose diameter is less than or equal to that of any previously proposed graph.

Ear-decompositions of matching-covered graphs

Abstract: We call a graphmatching-covered if every line belongs to a perfect matching. We study the technique of “ear-decompositions” of such graphs. We prove that a non-bipartite matching-covered graph containsK 4 orK 2⊕K 3 (the triangular prism). Using this result, we give new characterizations of those graphs whose matching and covering numbers are equal. We apply these results to the theory of τ-critical graphs.

Approximation algorithms for NP-complete problems on planar graphs

Abstract: This paper describes a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs. The strategy depends on decomposing a planar graph into subgraphs of a form we call k- outerplanar. For fixed k, the problems of interest are solvable optimally in linear time on k-outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum independent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least (k-1)/k optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k + 1)/k optimal. Taking k = c log log n or k = c log n, where n is the number of nodes and c is some constant, we get polynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approximation schemes includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k-outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable.

Some Problems on Dynamic/Periodic Graphs

Abstract: A dynamic graph is a (locally finite) infinite graph G = (V, E) in which the vertex set is V = {ip:i=1,…, n and p ∈ ℤ}, where ℤ is the set of integers, and the edge set has the following periodic property: (ip, jr) is an edge of E if and only if (ip+1, jr+1) is an edge of E. Dynamic graphs may model a wide range of periodic combinatorial optimization problems in workforce scheduling, vehicle routing, and production scheduling. Here we provide polynomial time algorithms for several elementary problems including the following: determining the connected components, determining the strongly connected components, determining an eulerian path if one exists, and determining a 2-coloring if one exists. (Here, the polynomial is in the finite number of bits needed to describe a dynamic graph.) In each case the problem on the dynamic graphs reduces to a related (but distinct) problem on a finite graph.

Constant-to-one and onto global maps of homomorphisms between strongly connected graphs

Abstract: The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

Approximation Algorithms for NP.Complete Problems on Planar Graphs (Preliminary Version

Abstract: ABSTRACf This paper describes a general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs. The strategy depends on decom­ posing a planar graph into subgraphs of a form we call k­ outerplanar. For fixed k, the problems of interest are solv­ able optimally in linear time on k -outerplanar graphs by dynamic programming. For general planar graphs, if the problem is a maximization problem, such as maximum in­ dependent set, this technique gives for each k a linear time algorithm that produces a solution whose size is at least (k -1)/k optimal. If the problem is a minimization problem, such as minimum vertex cover, it gives for each k a linear time algorithm that produces a solution whose size is at most (k+l)/k optimal. Taking k-cloglogn or k-clogn, where n is the number of nodes and c is some constant, we get po­ lynomial time approximation schemes, i.e. algorithms whose solution sizes converge toward optimal as n increases. The class of problems for which this approach provides approxi­ mation schemes includes maximum independent set, max­ imum tile salvage, partition into triangles, maximum H­ matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. For these and certain other problems, the proof of solvability on k -outerplanar graphs also enlarges the class of planar graphs for which the problems are known to be solvable.

Isomorphism of k-contractible graphs. A generalization of bounded valence and bounded genus

Abstract: A polynomial time isomorphism test for graphs called k-contractible graphs for fixed k is included. The class of k-contractible graphs includes the graphs of bounded valence and the graphs of bounded genus. The algorithm uses several new ideas including: (1) It removes portions of the graph and replaces them with groups which are used to keep track of the symmetries of these portions. (2) It maintains with each group a tower of equivalence relations which allow a decomposition of the group. These towers are called towers of Γk-actions.

Regular graphs with given girth pair

Abstract: The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. The existence of regular graphs with given degree and girth pair is proved and simple bounds for their smallest order are developed. Several infinite classes of such graphs are constructed and it is proved that two of these families consist of smallest graphs.

Extremal graph problems and graph products

Abstract: In this paper we consider only graphs without loops and multiple edges The product of two vertex disjoint graphs G 1 and G 2 is the graph obtained by joining each vertex of G 1 to each vertex of G 2 Given n and the sample graphs L 1,, L λ, we shall consider those graphs on n vertices which contain no L i, as a subgraph and have maximum number of edges under this condition These graphs will be called extremal graphs for the L i’s

Segment graphs, depth-first cycle bases, 3-connectivity, and planarity of graphs

Abstract: (1983). Segment graphs, depth-first cycle bases, 3-connectivity, and planarity of graphs. Linear and Multilinear Algebra: Vol. 13, No. 2, pp. 119-141.

Algebraic structures on graphs

Abstract: In this paper we consider directed graphs with algebraic structures: group-graphs, ringgraphs, involutorial graphs, affine graphs, graphs of morphisms between graphs, graphs of reduced paths of an involutorial graph, etc. We show also how several well-known algebraic constructions can be carried over to graphs. As a typical example we generalize the construction of the group of automorphisms of a set, by constructing a group-graph associated with any given graphΓ. It is the group-graph of reduced paths of the involutorial graph associated to the graph of automorphisms ofΓ.

Circulants and the Characterization of Vertex-Transitive Graphs.

Abstract: In this paper, we extend the notion of a circulant to a broader class of vertextransitive graphs, which we call multidimensional circulants This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive graph from any transitive subgroup of its automorphism group The approach also allows a short proof of Turner's theorem that every vertex-transitive graph on a prime number of nodes is a circulant

Construction of 4-Regular Graphs

Abstract: We show that the class of 4-regular, simple, connected graphs can be generated by three extensions from K 5 and by two extensions from K 5 and two other graphs of order 6 and 11. These constructions yield a proof of the existence of an edge-partition of a 4-regular graph in three linear forests.

A graph and its complement with specified properties. VI. Chromatic and achromatic numbers.

Abstract: is any connected graph of order 6 in List C.Of the 171 graphs which appear in these lists, 116 have x = x = 3. Inaddition to these the complements of the 51 graphs in List E areconnected graphs of order 7 through 9 with x = x = 3. And Corollary 2dimplies that there are many other graphs of order 7 through 9 with

On Trivalent Graphs Embedded in Twisted Honeycombs

Abstract: Publisher Summary This chapter discusses the trivalent graphs embedded in twisted honeycombs When centers of faces of a regular honeycomb {3,q,3} are connected to the midpoints of their edges, one obtains a 3-regular bipartite trivalent graph of girth 2q Twisted honeycombs {3,q,3} t likewise yield bipartite graphs, but in this case, graphs are only 2-regular The symmetry group of a twisted honeycomb contains no reflections and therefore, its right-handed and left-handed Petrie polygons may have different numbers of sides Hence, three consecutive edges of the corresponding bipartite graph may belong either to a right-handed or left-handed twisted polygon, making the graph 2-regular The edges of the graph can be 3-colored as to provide the Cayley diagram for a group of order 192

A generalisation of a recent characterisation of planar graphs

Abstract: Planar graphs have recently been characterised as those which have no strict elegant odd ring of circuits. Here we generalise that result by showing that its dual yields a theorem that is valid for all graphs.