Irreducible graphs

The fact that there is exactly one cubic graph irreducibly non-representable on a plane was demonstrated by C. Kuratowski in 1930. This paper gives bounds for the number of nodes in a cubic graph irreducibly non-representable on a surface S in terms of the number of nodes of subgraphs. An upper bound (866) is given for the number of nodes in a cubic graph irreducibly non-representable on a projective plane.     

On maximum bipartite subgraphs

In this paper are investigated maximum bipartite subgraphs of graphs, i.e., bipartite subgraphs with a maximum number of edges. Such subgraphs are characterized and a criterion is given for a subgraph to be a unique maximum bipartite subgraph of a given graph. In particular maximum bipartite subgraphs of cubic graphs are investigated. It is shown that cubic graphs can be built up from five building stones (called elementary paths). Finally the investigation of a special class of cubic graphs yields a theorem which characterizes the Petersen graph and the dodecahedron graph by means of their maximum bipartite subgraphs.     

Parity and disparity subgraphs

A parity subgraph of a graph is a spanning subgraph such that the degrees of each vertex have the same parity in both the subgraph and the original graph. Known results include that every graph has an odd number of minimal parity subgraphs. Define a disparity subgraph to be a spanning subgraph such that each vertex has degrees of opposite parities in the subgraph and the original graph. (Only graphs with all even-order components can have disparity subgraphs). Every even-order spanning tree contains both a unique parity subgraph and a unique disparity subgraph. Moreover, every minimal disparity subgraph is shown to be paired by sharing a spanning tree with an odd number of minimal parity subgraphs, and every minimal parity subgraph is similarly paired with either one or an even number of minimal disparity subgraphs.     

Matching shape graphs

Graphs are important structures to model complex relationships such as chemical compounds, proteins, geometric or hierarchical parts, and XML documents. Given a query graph, indexing has become a necessity to retrieve similar graphs quickly from large databases. We propose a novel technique for indexing databases, whose entries can be represented as graph structures. Our method starts by representing the topological structure of a graph as well as that of its subgraphs as vectors in which the components correspond to the sorted laplacian eigenvalues of the graph or subgraphs. By doing a nearest neighbor search around the query spectra, similar but not necessarily isomorphic graphs are retrieved. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The structure of 1-planar graphs

A graph is called 1-planar if it can be drawn in the plane so that each its edge is crossed by at most one other edge. In the paper, we study the existence of subgraphs of bounded degrees in 1-planar graphs. It is shown that each 1-planar graph contains a vertex of degree at most 7; we also prove that each 3-connected 1-planar graph contains an edge with both endvertices of degrees at most 20, and we present similar results concerning bigger structures in 1-planar graphs with additional constraints.     

A distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} and its automorphisms

Possible orders and subgraphs of the fixed points of a distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} are found. It is shown that such a graph is not vertex-transitive.     

Crossing-critical edges and Kuratowski subgraphs of a graph

An edge e of a graph G is said to be crossing-critical if cr(G ? e) < cr(G), where cr(G) denotes the crossing number of G on the plane. It is proved that any crossing-critical edge e of a graph G for which cr(G ? e) ≤ 1 belongs to a subdivision of K₅ or K_{3, 3}, the Kuratowski subgraphs of G. Further, as regards crossing-critical edges e of G for which cr(G ? e) ≥ 5, it is shown that the properties of “being a crossing-critical edge of G” and “being contained in a Kuratowski subgraph of G” are independent.     

The theorem on planar graphs

In the late 1920s several mathematicians were on the verge of discovering a theorem for characterizing planar graphs. The proof of such a theorem was published in 1930 by Kazimierz Kuratowski, and soon thereafter the theorem was referred to as the Kuratowski Theorem. It has since become the most frequently cited result in graph theory. Recently, the name of Pontryagin has been coupled with that of Kuratowski when identifying this result. The events related to this development are examined with the object of determining to whom and in what proportion the credit should be given for the discovery of this theorem.     

Optimal implementation of selectors in a data-base

The records of a data base can be accessed from other records or from a set of data items (inverted access, primary and secondary index of IMS, search keys of CODASYL etc.) which we call selectors. The implementation of this selectors can use different techniques as hash coding, inverted lists or hierarchical index (indexed sequential, B-trees etc…) We consider here the last one and we search for a given set of selectors an optimal index structure. We show how this problem can be put as the search of an optimal rooted tree among the partial subgraphs of a given graph G (this problem is known in graph theory as Steiner problem) and we give several properties which allow the graph G to be notabily reduced. Then we present a branch and bound algorithm to solve this problem.     

Isomorphic factorizations of complete equipartite graphs

A complete equipartite graph is a complete multipartite graph such that the color classes are equinumerous. An isomorphic factorization of a graph is a partition of the line set into disjoint isomorphic parts. It is shown that an isomorphic factorization of the complete equipartite graph into t isomorphic subgraphs exists whenever the total number of lines is divisible by t.     

Optimization of dialectical outcomes in dialogical argumentation

When informal arguments are presented, there may be imprecision in the language used, and so the audience may be uncertain as to the structure of the argument graph as intended by the presenter of the arguments. For a presenter of arguments, it is useful to know the audience's argument graph, but the presenter may be uncertain as to the structure of it. To model the uncertainty as to the structure of the argument graph in situations such as these, we can use probabilistic argument graphs. The set of subgraphs of an argument graph is a sample space. A probability value is assigned to each subgraph such that the sum is 1, thereby reflecting the uncertainty over which is the actual subgraph. We can then determine the probability that a particular set of arguments is included or excluded from an extension according to a particular Dung semantics. We represent and reason with extensions from a graph and from its subgraphs, using a logic of dialectical outcomes that we present. We harness this to define the notion of an argumentation lottery, which can be used by the audience to determine the expected utility of a debate, and can be used by the presenter to decide which arguments to present by choosing those that maximize expected utility. We investigate some of the options for using argumentation lotteries, and provide a computational evaluation.     

A result on Hamiltonian line graphs involving restrictions on induced subgraphs

It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph.     

Orderly algorithms for generating restricted classes of graphs

Orderly algorithms for the generation of exhaustive lists of nonisomorphic graphs are discussed. The existence of orderly methods to generate the graphs with a given subgraph and without a given subgraph is established. This method can be used to list all the nonisomorphic subgraphs of a given graph, as well as to produce catalogs of Hamiltonian graphs, pancyclic graphs, degree-constrained graphs, and other classes. A generalization of this method is given that can be used to generate lists of graphs with given girth, planar graphs, k-colorable graphs, and k-connected graphs, for example. Finally, these observations are employed to generate restricted classes of digraphs, notably acyclic digraphs and poset digraphs. The generation of poset digraphs is shown to supply a practical orderly method for producing a catalog of lattices. Similar observations concerning vertex addition generation methods allow one to improve on existing methods for the generation of catalog of interval and circle graphs.     

The edge reconstruction hypothesis is true for graphs with more than n · log2n edges

It is shown that a graph with n vertices and more than n · log₂n edges can be uniquely reconstructed from its edge-deleted subgraphs.     

Complement reducible graphs

In this paper we study the family of graphs which can be reduced to single vertices by recursively complementing all connected subgraphs. It is shown that these graphs can be uniquely represented by a tree where the leaves of the tree correspond to the vertices of the graph. From this tree representation we derive many new structural and algorithmic properties. Furthermore, it is shown that these graphs have arisen independently in various diverse areas of mathematics.     

Optical grooming with grooming ratio eight

Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum possible drop cost is determined for grooming ratio 8, and this cost is shown to be realized with six exceptions, and 37 possible exceptions, the largest being 105.     

Alternating cycles in edge-partitioned graphs

It is shown that if the edges of a 2-connected graph G are partitioned into two classes so that every vertex is incident with edges from both classes, then G has an alternating cycle. The connectivity assumption can be dropped if both subgraphs resulting from the partition are regular, or have only vertices of odd degree.     

Information flow in graphs

A formula is obtained for the global flow of information in a discrete Markov system which is defined on a locally finite graph. It is shown that the information flow is related to a random walk on the lattice of finite subgraphs of the graph.