Motives of melonic graphs

The metric representation of a vertex v of a graph G is a finite vector representing distances of v with respect to vertices of some ordered subset S⊆V (G). If no suitable subset of S provides separate representations for each vertex of V(G), then the set S is referred to as a minimal resolving set. The metric dimension of G is the cardinality of the smallest (with respect to its cardinality) minimal resolving set. A resolving set S is secure if for any v∈V–S, there exists x∈S such that (S–{x})∪{v} is a resolving set. For various classes of graphs, the value of the secure resolving number is determined and defined. The secure metric dimension of the graph classes is being studied in this work. The results show that different graph families have different metric dimensions.

Accordion graphs: Hamiltonicity, matchings and isomorphism with quartic circulants

Coloring and domination in graphs are two well explored areas of research in graph theory. Blending these notions, the dominator coloring of graphs was introduced in the literature; following which several variants of domination related coloring patterns have been defined and studied, based on different types of coloring and domination in graphs. A vertex coloring of a graph that demands the existence of a path in which every internal vertex between two vertices has a unique color is called a rainbow vertex coloring of the graph. In this article, we investigate the rainbow dominator coloring of graphs; a vertex coloring that combines the concepts of rainbow vertex coloring and dominator coloring of graphs. We discuss some properties of the rainbow dominator coloring of graphs and determine the rainbow dominator chromatic number of certain classes of graphs and their complements.

A thrackle is a drawing of a graph in which any two vertex-disjoint edges cross exactly once and incident edges do not cross. A graph that has a thrackle drawing is thracklable. In the past three decades, mathematicians investigated a number of variations of thrackles by relaxing or changing the required number of crossings on edges or restricting the placement of vertices on the surface (e.g., restricting to outerdrawings, in which vertices are restricted to a boundary curve of the surface). A graph is superthracklable if it can be drawn so that any two edges cross exactly once. We provide a simple proof for the following: a graph can be drawn on the plane so that every two edges cross an odd number of times if and only if it is superthracklable on the plane. A graph is outersuperthracklable if it has a drawing on a disc in which every vertex is on the boundary of the disc and every pair of edges cross exactly once. We introduce some natural generalisations of outersuperthracklable graphs and we show that these classes of graphs are equivalent. Lastly, using the Hanani-Tutte characterisation of planar graphs we show that for any surface $\Sigma$, there is a relationship between the class of graphs that are not embeddable on $\Sigma$ and the class of graphs that are not superthracklable with respect to $\Sigma$. More specifically, we show how to construct, from any forbidden minor $G$ for embeddability in $\Sigma$, two infinite families of graphs that are not superthracklable with respect to $\Sigma$. This sheds further light on the relationship between some of Archdeacon and Stor's forbidden configurations for superthrackles and Kuratowski's forbidden minors for planarity.

We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be \(\Omega(\sqrt{n})\) even for planar graphs [11] due to a simple topological obstruction and a major focus, following earlier work [14], has been understanding the gap if some constant congestion is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [19,5]. In general graphs, recent work has shown the gap to be O(polylog(n)) [8,9] with congestion 2. Moreover, the gap is Ω(logΩ(c) n) in general graphs with congestion c for any constant c ≥ 1 [1].It is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of “nice” graphs is minor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant-factor constant-congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution’s value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to k-sums of graphs in some graph family, as long as the graph family has a constant-factor constant-congestion bound. We then use this to show that such bounds hold for the class of k-sums of bounded genus graphs.KeywordsPlanar GraphTree DecompositionGraph FamilyDisjoint Path ProblemSeries Parallel GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

String graphs of k-bend paths on a grid

Letter graphs and modular decomposition

Graph classification is one of the most important research fields in data mining nowadays and many algorithms have been proposed to address this issue. In practice, labeling large or even medium-size graphs is a hard task and we need experts to do so. The biggest challenge in graph classification is extracting a set of proper features from graphs. Since graphs are represented by a complex data structure, this issue has been dealt with for a long time. Previous methods focused on extracting features from a certain class in a dataset. In this paper we propose a new feature selection method that extracts features from each graph rather than extracting them from a certain class in the dataset. We extract only frequent subgraphs as features. These subgraphs are chosen according to their number of occurrences in a graph. Moreover, we proposed a new formula which calculates the minimum number of occurrences required for a subgraph to be considered as frequent.We experimented on five real datasets and reached 7-17% higher accuracy than previously proposed methods.

Constructions of sensitive graphs which are not strongly sensitive

On S-Magic Graphs

We propose a class of hierarchical graphs equipped with a simple algebraic syntax as a convenient way to describe configurations in languages with inherently hierarchical features such as sessions, fault-handling scopes or transactions. The graph syntax can be seen as an intermediate representation language, that facilitates the encoding of structured specifications and, in particular, of process calculi, since it provides primitives for nesting, name restriction and parallel composition. The syntax is based on an algebraic presentation that faithfully characterises families of hierarchical graphs, meaning that each term of the language uniquely identifies an equivalence class of graphs (modulo graph isomorphism). Proving soundness and completeness of an encoding (i.e. proving that structurally equivalent processes aremapped to isomorphic graphs) is then facilitated and can be done by structural induction. Summing up, the graph syntax facilitates the definition of faithful encodings, yet allowing a precise visual representation. We illustrate our work with an application to a work-flow language and a service-oriented calculus.

We examine the generic local and global rigidity of various graphs in ℝd . Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity, and hypothesized that they were sufficient in all dimensions. We analyze two classes of graphs that satisfy Hendrickson’s conditions for generic global rigidity, yet fail to be generically globally rigid. We find a large family of bipartite graphs for d>3, and we define a construction that generates infinitely many graphs in ℝ5. Finally, we state some conjectures for further exploration.

In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to Kemeny's constant: in particular, we examine graphs with a cut-vertex at which at least two branches are paths, regarding whether the insertion of a particular edge into a graph results in an increase of Kemeny's constant. We provide several tools for identifying such an edge in a family of graphs and for analysing asymptotic behaviour of the family regarding the tendency to have that edge; and classes of particular graphs are given as examples. Furthermore, asymptotic behaviours of families of trees are described.

Orbits of graphs under local complementation (LC) and edge local complementation (ELC) have been studied in several different contexts. For instance, there are connections between orbits of graphs and error-correcting codes. We define a new graph class, ELC-preserved graphs, comprising all graphs that have an ELC orbit of size one. Through an exhaustive search, we find all ELC-preserved graphs of order up to 12 and all ELC-preserved bipartite graphs of order up to 16. We provide general recursive constructions for infinite families of ELC-preserved graphs, and show that all known ELC-preserved graphs arise from these constructions or can be obtained from Hamming codes. We also prove that certain pairs of ELC-preserved graphs are LC equivalent. We define ELC-preserved codes as binary linear codes corresponding to bipartite ELC-preserved graphs, and study the parameters of such codes.

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].