ДОМИНАНТНО-ТРЕУГОЛЬНЫЕ ГРАФЫ И ГРАФЫ ВЕРХНИХ ГРАНИЦ

In this article we introduce and study a proper subclass of triangle graphs, namely, the domination triangle graphs. A graph G is called a domination triangle graph if for every minimal dominating set D of G and any adjacent vertices u and v not contained in D, there exists a vertex w є D such that the set {u, v, w} induces a triangle in G. We propose a number of characterizations of domination triangle graphs which show in particular that this class of graphs coincides with a well-known class of upper bound graphs and a class of irredundance triangle graphs. We establish the computational complexity and the complexity of approximation for some independence and domination-related graph-theoretical parameters within domination triangle graphs.В работе вводится и изучается собственный подкласс треугольных графов, а именно доминантно-треугольные графы. Граф G называется доминантно-треугольным, если для любого минимального доминирующего множества D графа G и любых смежных вершин u и v, не входящих в D, существует вершина w є D, одновременно смежная с u и v, т. е. множество {u, v, w} порождает треугольник в графе G. Получен ряд характеризаций класса доминантно-треугольных графов, которые, в частности, указывают на совпадение этого класса графов с хорошо известным классом графов верхних границ и классом ирридантно-треугольных графов. Установлена вычислительная сложность и сложность аппроксимации в классе доминантно-треугольных графов некоторых теоретико-графовых параметров, родственных классическим числам независимости и доминирования

ХАРАКТЕРИЗАЦИЯ 1-ТРЕУГОЛЬНЫХ ГРАФОВ

A graph is called 1-triangle if for each maximal independent set I, each edge of this graph with both end vertices not belonging to I forms exactly one triangle with a vertex from the set I. We have obtained a structural characterization of 1-triangle graphs which implies a polynomial time recognition algorithm for this class of graphs.Граф называется 1-треугольным, если для любого максимального независимого множества I этого графа каждое ребро графа, не инцидентное ни одной вершине из I, образует единственный треугольник с вершиной из множества I. В работе получена структурная характеризация класса 1-треугольных графов, которая влечёт полиномиальный алгоритм их распознавания

Biological Effect of Continuous, Quasi-Continuous and Pulsed Laser Radiation

In this work, for the first time, comparative studies of biological activity of low intensity continuous, quasi-continuous and pulsed laser radiation of nano- and picosecond time ranges with the same average power density are carried out. It is shown, that, despite the significant differences in peak values of intensity of acting factor, both continuous and quasi-continuous radiation and radiation of nano- and picosecond ranges are able to have both stimulating and inhibiting effects on all investigated parameters of functional activity of biological systems in a certain range of dose rates. The ability of laser radiation of near infra-red spectral region (800 - 1340 nm) located out the absorption bands of main chromophores of cells to have regulatory effect on biochemical processes that control the hatching of branchiopod crustaceans Artemia salina L. upon irradiation of their cysts is revealed. The role of molecular oxygen and water as acceptors of laser radiation is discussed. Keywords: Low intensity laser radiation, Laser activation, Biological activity, Zooplankton Artemia salina L., Sturgeon sperm

Hamiltonian properties of triangular grid graphs

Available online at www.sciencedirect.comA triangular grid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional triangular grid. In 2000, Reay and Zamfirescu showed that all 2-connected, linearly-convex triangular grid graphs (with the exception of one of them) are hamiltonian. The only exception is a graph D which is the linearly-convex hull of the Star of David. We extend this result to a wider class of locally connected triangular grid graphs. Namely, we prove that all connected, locally connected triangular grid graphs (with the same exception of graph D) are hamiltonian. Moreover, we present a sufficient condition for a connected graph to be fully cycle extendable. We also show that the problem HAMILTONIAN CYCLE is NP-complete for triangular grid graphs

Approximability results for the maximum and minimum maximal induced matching problems

An induced matching M of a graph G is a set of pairwise non-adjacent edges such that their end-vertices induce a 1-regular subgraph. An induced matching M is maximal if no other induced matching contains M. The MINIMUM MAXIMAL INDUCED MATCHING problem asks for a minimum maximal induced matching in a given graph. The problem is known to be NP-complete. Here we show that, if P 6= NP, for any " > 0, this problem cannot be approximated within a factor of n1−" in polynomial time, where n denotes the number of vertices in the input graph. The result holds even if the graph in question is restricted to being bipartite. For the related problem of finding an induced matching of maximum size (MAXIMUM INDUCED MATCHING), it is shown that, if P 6= NP, for any " > 0, the problem cannot be approximated within a factor of n1/2−" in polynomial time. Moreover, we show that MAXIMUM INDUCED MATCHING is NP-complete for planar line graphs of planar bipartite graphs

Hamiltonian properties of locally connnected graphs with bounded vertex degree

We consider the existence of hamiltonian cycles for locally connected graphs with a bounded vertex degree. For a graph G, let ¢(G) and ±(G) denote the maximum and minimum vertex degrees, respectively. We explicitly describe all connected, locally connected graphs with ¢(G) 6 4. We show that every connected, locally connected graph with ¢(G) = 5 and ±(G) > 3 is fully cycle extendable which extends the results of P.B. Kikust (Latvian Math. Annual 16 (1975) 33{38) and G.R.T. Hendry (J. Graph Theory 13 (1989) 257{260) on fully cycle extendability of connected, locally connected graphs with the maximum vertex degree bounded by 5. Furthermore, we prove that problem Hamilton Cycle for locally connected graphs with ¢(G) 6 7 is NP-complete. 2000 Mathematics Subject Classi¯cation: 05C38 (05C45, 68Q25)

Computational complexity of maximum distance-(k, l) matchings in graphs

Секция 10. Теоретическая информатикаIn this paper, we introduce the concept of a distance-(k, l) matching of a graph, which is a subset of edges of this graph such that the number of intermediate edges in the shortest path between any two edges of this set lies between k and l. We prove that the problem MAXIMUM DISTANCE-(k, l) MATCHING, which asks whether a graph contains a distance-(k, l) matching of size exceeding a given number, is NP-complete for arbitrary given or variable k and l, and that the weighted variant of this problem is strongly NP-complete even for bipartite graphs. We also present several upper bounds on the size of a maximum distance-(k, l) matching

Setting the scene

Fungi are a keystone component of all ecosystems on earth and have shaped the structure and functioning of nature for eons. Their body is made up of an interwoven mass of threadlike filaments, individually called hyphae and altogether known as mycelium. When fungi form spore-forming structures, the so-called mushrooms, these are also built up of hyphae. The three known trophic groups of fungi have had a fundamental functional diversity in the development of life as we know it. The so-called saprotrophs use the complex dead materials of plants, animals, and microorganisms, including other fungi, as their source of energy—playing a crucial role in nutrient recycling in nature. Mycorrhizal fungi, which establish symbiotic relationships with the roots of plants, have gone hand in hand with plants in the colonization of life on land environments since more than 400 million years. Ectomycorrhizal fungi, in particular, are currently essential in maintaining forest masses worldwide since they establish mutualistic relationships with trees and shrubs mainly. According to the recently named Read’s rule, they dominate ecosystems with low mineralization rates in high-latitude regions with cold and dry climates. Furthermore, they have also been considered the earth’s natural internet, or ‘wood wide web’, because they connect plants and enable them to share nutrients, water, and signal compounds among individuals and species. These networks also help stocking carbon in organic forms from the atmosphere, contributing to climate regulation. Some fungi also establish parasitic relationships with plants and animals, working as an evolutionary force and a selection pressure factor of paramount importance in these groups of living organisms. On top of this ecological and evolutionary relevance, the reproductive structures of fungi, the mushrooms, have shapes and colours that have always fascinated humans. Since early human history, mushrooms have also been an important source of food, medicine, and ceremonial use, all around the world. They also cause death or disease, since deadly and poisonous species exist. Nowadays, they are an important source for the search of new antibiotics, enzymes with industrial use, bioremediation, biofuels, cosmetics, inks, and dyes. In this introductory chapter, we will first describe some remarkable ecological facts related to mushrooms, including members of the three trophic groups previously mentioned. Then, we will provide evidences of the ancient relationships between mushrooms and humans; and, finally, we will analyse the relationships between mushrooms, humans, and nature in different parts of the world, describing and illustrating different realities in five continents