On a programming language for graph algorithms

An algorithmic language, GRAAL, is defined, as an extension of ALGOL 60 (Revised), for describing and implementing graph algorithms of the type arising in applications. It is based on a set algebraic model of graph theory which defines the graph structure in terms of user specified morphisms between certain set algebraic structures over the node and arc set. Several examples of graph algorithms written in GRAAL are included.This work was in part supported by Grant GJ-1067 from the U.S. National Science Foundation and Grant NGL-21-002-008 from the U.S. National Aeronautics and Space Administration.     

The common structure graph: Common structural features of a set of graphs

A method is proposed for abstracting the common features of a set of graphs. It is based on the graph homomorphisms that a set of graphs share. A semilattice structure is imposed on the partial order of graph homomorphisms of a set of graphs. The ‘common structure graphs’ are defined in relation to this semilattice.     

Characterizations of bipartite and Eulerian partial duals of ribbon graphs

Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of semi-crossing directions of its medial graph. Plane graphs are ribbon graphs with genus 0. In this paper, by introducing the notion of modified medial graphs and using their all-crossing directions, we first extend Huggett and Moffatt’s result from plane graphs to ribbon graphs. Then we characterize all Eulerian partial duals of any ribbon graph in terms of crossing-total directions of its medial graph, which are simpler than semi-crossing directions.     

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth of a decomposition tree of $G$ in terms of basic perfect graphs.     

Almost self-centered graphs

The center of a graph is the set of vertices with minimum eccentricity. Graphs in which all vertices are central are called self-centered graphs. In this paper almost self-centered (ASC) graphs are introduced as the graphs with exactly two non-central vertices. The block structure of these graphs is described and constructions for generating such graphs are proposed. Embeddings of arbitrary graphs into ASC graphs are studied. In particular it is shown that any graph can be embedded into an ASC graph of prescribed radius. Embeddings into ASC graphs of radius two are studied in more detail. ASC index of a graph G is introduced as the smallest number of vertices needed to add to G such that G is an induced subgraph of an ASC graph.     

Power domination in graphs

The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 1998, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number γ_P(G) of a graph G is the minimum cardinality of a power dominating set of G. In this paper, we present upper bounds on the power domination number for a connected graph with at least three vertices and a connected claw-free cubic graph in terms of their order. The extremal graphs attaining the upper bounds are also characterized.     

Hereditary Efficiently Dominatable Graphs

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the graph's vertex set. We introduce graphs that are hereditary efficiently dominatable in that sense that every induced subgraph of the graph contains an efficient dominating set. We prove a decomposition theorem for (bull, fork, C₄)‐free graphs, based on which we characterize, in terms of forbidden induced subgraphs, the class of hereditary efficiently dominatable graphs. We also give a decomposition theorem for hereditary efficiently dominatable graphs and examine some algorithmic aspects of such graphs. In particular, we give a polynomial time algorithm for finding an efficient dominating set (if one exists) in a class of graphs properly containing the class of hereditary efficiently dominatable graphs by reducing the problem to the maximum weight independent set problem in claw‐free graphs.     

Witness (Delaunay) graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations.We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.     

Subgraphs of hypercubes and subdiagrams of boolean lattices

In this paper we extensively treat the following problems: When is a given graph a subgraph (resp. induced subgraph) of a hypercube and when is an ordered set a subdiagram (resp. induced subdiagram) of a Boolean lattice? We present characterizations for that in terms of suitable edge-colorings of the graphs and, for ordered sets, of their covering graphs.     

Semi‐transitive graphs

In this paper, we first consider graphs allowing symmetry groups which act transitively on edges but not on darts (directed edges). We see that there are two ways in which this can happen and we introduce the terms bi‐transitive and semi‐transitive to describe them. We examine the elementary implications of each condition and consider families of examples; primary among these are the semi‐transitive spider‐graphs PS(k,N;r) and MPS(k,N;r). We show how a product operation can be used to produce larger graphs of each type from smaller ones. We introduce the alternet of a directed graph. This links the two conditions, for each alternet of a semi‐transitive graph (if it has more than one) is a bi‐transitive graph. We show how the alternets can be used to understand the structure of a semi‐transitive graph, and that the action of the group on the set of alternets can be an interesting structure in its own right. We use alternets to define the attachment number of the graph, and the important special cases of tightly attached and loosely attached graphs. In the case of tightly attached graphs, we show an addressing scheme to describe the graph with coordinates. Finally, we use the addressing scheme to complete the classification of tightly attached semi‐transitive graphs of degree 4 begun by Marus?ic? and Praeger. This classification shows that nearly all such graphs are spider‐graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 1–27, 2004     

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K_2,2 and its complement, all complete graphs and no other graphs.     

Multithreshold graphs

Multithreshold graphs are defined in terms of a finite sequence of real thresholds that break the real line into a set of regions, alternating between NO and YES. If real ranks can be assigned to the vertices of a graph in such a way that two vertices are adjacent iff the sum of their ranks lies in a YES region, then that graph is a multithreshold graph with respect to the given set of thresholds. If a graph can be represented with k or fewer thresholds, then it is k-threshold. The case of one threshold is the classical case introduced by Chvátal and Hammer. In this paper, we show for every graph G, there is a k such that G is k-threshold, and we exhibit graphs for which the required number of thresholds is linear in the order of the graph.     

Paired-domination of Cartesian products of graphs and rainbow domination

The most famous open problem involving domination in graphs is Vizing's conjecture which states the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. We investigate a similar problem for paired-domination, and obtain a lower bound in terms of product of domination number of one factor and 3-packing of the other factor. Some results are obtained by applying a new graph invariant called rainbow domination.     

The Laplacian spectra of graphs with a tree structure

Trees are very common in the theory and applications of combinatorics. In this article, we consider graphs whose underlying structure is a tree, except that its vertices are graphs in their own right and where adjacent graphs (vertices) are linked by taking their join. We study the spectral properties of the Laplacian matrices of such graphs. It turns out that in order to capture known spectral properties of the Laplacian matrices of trees, it is necessary to consider the Laplacians of vertex-weighted graphs. We focus on the second smallest eigenvalue of such Laplacians and on the properties of their corresponding eigenvector. We characterize the second smallest eigenvalue in terms of the Perron branches of a tree. Finally, we show that our results are applicable to advancing the solution to the problem of whether there exists a graph on n vertices whose Laplacian has the integer eigenvalues 0, 1, …, n ? 1.     

Volume entropy,weighted girths and stable balls on graphs

We prove new isoperimetric inequalities on graphs involving quantities linked with concepts from differential geometry. First, we bound from above the product of the volume entropy (defined as the log of the exponential growth rate of the universal cover) and the girth of weighted graphs in terms of their cyclomatic number. In a second part, we study a natural polyhedron associated to a weighted graph: the stable ball. In particular, we relate the volume of this polyhedron, the weight of the graph and its cyclomatic number. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 291–305, 2007     

Coincidence of the sets of minimal and irreducible join graphs over primary structure of algebraic Bayesian networks

An algebraic Bayesian network (ABN) is a probabilistic-logic graphical model of bases of knowledge patterns with uncertainty. A primary structure of an ABN is a set of knowledge patterns, that are ideals of conjunctions of positive literals except the empty conjunction endowed with scalar or interval probability estimates. A secondary ABN structure is represented by a graph constructed over the primary structure, which is called a join graph. From the point of view of learning of a global ABN structure, of interest are join graphs with the minimum number of edges and irreducible join graphs. A theorem on the coincidence of the sets of minimal and irreducible join graphs over the same primary structure is proved. A greedy algorithm constructing an arbitrary minimal join graph from a given primary structure is described. A theorem expressing the number of edges in a minimal join graph as the sum of the ranks of the incidence matrices of strong restrictions of a maximal join graph minus the number of significant weights is stated and proved. A generalized graph of maximal knowledge patterns (GGMKP) is a graph with the same vertex set as the join graph which is not subject to any constraints concerning the possibility of joining two vertices by an edge. It is proved that the pair consisting of the edge set of a maximal GGMKP and the set of all subsets of this graph such that the subtraction of any such subset from the maximal GGMKP yields an edge of the join graph on the same vertex set is a matroid.     

Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ₁. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ₁ eigenvalue for connected graphs.     

Graphs with Branchwidth at Most Three

In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three if and only if it has treewidth at most three and does not contain the three-dimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set for the class of graphs with branchwidth at most three. Moreover, we give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that verifies if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition.     

Two Bounds for the Domination Number of a Graph

Upper and lower bounds are obtained for the domination numberof a graph, by means of a lemma involving the concept of a minimumdominating set of vertices. Although these results are obtainedexplicitly for graphs, there are analogous results in the theoryof directed graphs.     

A natural graph of finite fields distinguishing between models

We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields. Certain basic notions in finite field theory have interpretations in terms of standard graph properties. We show that the graphs are connected and provide an estimate of their diameter. An accidental graph isomorphism is uncovered and proved. The smallest non-trivial Laplace eigenvalue is given some attention, in particular for a specific family of 8-regular graphs showing that it is not an expander. We introduce a regular covering graph and show that it is connected if and only if the root is primitive.