Andrej Taranenko

Fast Recognition of Fibonacci Cubes

Abstract Fibonacci cubes are induced subgraphs of hypercubes based on Fibonacci strings. They were introduced to represent interconnection networks as an alternative to the hypercube networks. We derive a characterization of Fibonacci cubes founded on the concept of resonance graphs. The characterization is the basis for an algorithm which recognizes these graphs in O(mlog n) time.

On the k-path vertex cover of some graph products

A subset S of vertices of a graph G is called a k-path vertex cover if every path of orderk inG contains at least one vertex fromS. Denote by k(G) the minimum cardinality of a k-path vertex cover in G. In this paper improved lower and upper bounds for k of the Cartesian and the direct product of paths are derived. It is shown that for 3 those bounds are tight. For the lexicographic product bounds are presented for k, moreover 2 and 3 are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.

The partition dimension of strong product graphs and Cartesian product graphs

Abstract Let G = ( V , E ) be a connected graph. The distance between two vertices u , v ∈ V , denoted by d ( u , v ) , is the length of a shortest u , v -path in G . The distance between a vertex v ∈ V and a subset P ⊂ V is defined as min { d ( v , x ) : x ∈ P } , and it is denoted by d ( v , P ) . An ordered partition { P 1 , P 2 , … , P t } of vertices of a graph G , is a resolving partition of G , if all the distance vectors ( d ( v , P 1 ) , d ( v , P 2 ) , … , d ( v , P t ) ) are different. The partition dimension of G is the minimum number of sets in any resolving partition of G . In this article we study the partition dimension of strong product graphs and Cartesian product graphs. Specifically, we prove that the partition dimension of the strong product of graphs is bounded below by four and above by the product of the partition dimensions of the factor graphs. Also, we give the exact value of the partition dimension of strong product graphs when one factor is a complete graph and the other one is a path or a cycle. For the case of Cartesian product graphs, we show that its partition dimension is less than or equal to the sum of the partition dimensions of the factor graphs minus one. Moreover, we obtain an upper bound on the partition dimension of Cartesian product graphs, when one factor is a complete graph.

Maximum cardinality resonant sets and maximal alternating sets of hexagonal systems

It is shown that the Clar number can be arbitrarily larger than the cardinality of a maximal alternating set. In particular, a maximal alternating set of a hexagonal system need not contain a maximum cardinality resonant set, thus disproving a previously stated conjecture. It is known that maximum cardinality resonant sets and maximal alternating sets are canonical, but the proofs of these two theorems are analogous and lengthy. A new conjecture is proposed and it is shown that the validity of the conjecture allows short proofs of the aforementioned two results. The conjecture holds for catacondensed hexagonal systems and for all normal hexagonal systems up to ten hexagons. Also, it is shown that the Fries number can be arbitrarily larger than the Clar number.

Characterization of reducible hexagons and fast decomposition of elementary benzenoid graphs

A benzenoid graph is a finite connected plane graph with no cut vertices in which every interior region is bounded by a regular hexagon of a side length one. A benzenoid graph G is elementary if every edge belongs to a 1-factor of G. A hexagon h of an elementary benzenoid graph is reducible, if the removal of boundary edges and vertices of h results in an elementary benzenoid graph. We characterize the reducible hexagons of an elementary benzenoid graph. The characterization is the basis for an algorithm which finds the sequence of reducible hexagons that decompose a graph of this class in O(n^2) time. Moreover, we present an algorithm which decomposes an elementary benzenoid graph with at most one pericondensed component in linear time.

An elitist genetic algorithm for the maximum independent set problem

Genetic algorithms are a computational paradigm belonging to the class of optimization techniques known as evolutionary computation. They have been implemented successfully to solve many difficult optimization problems. We have developed a new genetic algorithm for the maximum independent set problem based on the elitist strategy. The algorithm presented is tested on the so-called DIMACS benchmark graphs. The effectiveness of the algorithm is very satisfactory since it outperforms in most cases the genetic algorithms for the maximum independent set problem reported in the literature.

Mixed metric dimension of graphs

Let G=(V,E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges) x, y ∈ E ∪ V if dG(w, x) ≠ dG(w, y). A set S of vertices in a connected graph G is a mixed metric generator for G if every two distinct elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by dimm(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.

1-FACTORS AND CHARACTERIZATION OF REDUCIBLE FACES OF PLANE ELEMENTARY BIPARTITE GRAPHS

As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekule structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the re- moval of the internal vertices and edges of the path that is the intersection of f and the outer cycle of G results in an elementary graph. We characterize the reducible faces of a plane elementary bipartite graph. This result gen- eralizes the characterization of reducible faces of an elementary benzenoid graph.

L(2,1)-coloring of the Fibonacci cubes

An L(2, l)-coloring of a graph G is an assignment of labels from {0,1,..., A} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart. The X-number X(G) of G is the minimum value A such that G admits an L(2,1)-coloring. It is well known that the problem of determining the X-number is NP-hard. The Fibonacci cube network was recently proposed as an alternative to the hypercube network. Three different evolutionary algorithms are presented to find optimal or near optimal L(2,1)-coloring of the Fibonacci cubes. The algorithms are compared with the Petford-Welsh probabilistic algorithm