On perfect and unique maximum independent sets in graphs

summary:A perfect independent set $I$ of a graph $G$ is defined to be an independent set with the property that any vertex not in $I$ has at least two neighbors in $I$. For a nonnegative integer $k$, a subset $I$ of the vertex set $V(G)$ of a graph $G$ is said to be $k$-independent, if $I$ is independent and every independent subset $I^{\prime }$ of $G$ with $|I^{\prime }|\ge |I|-(k-1)$ is a subset of $I$. A set $I$ of vertices of $G$ is a super $k$-independent set of $G$ if $I$ is $k$-independent in the graph $G[I,V(G)-I]$, where $G[I,V(G)-I]$ is the bipartite graph obtained from $G$ by deleting all edges which are not incident with vertices of $I$. It is easy to see that a set $I$ is $0$-independent if and only if it is a maximum independent set and 1-independent if and only if it is a unique maximum independent set of $G$. In this paper we mainly investigate connections between perfect independent sets and $k$-independent as well as super $k$-independent sets for $k=0$ and $k=1$

Hamiltonian paths containing a given arc, in almost regular bipartite tournaments

AbstractA tournament is an orientation of a complete graph, and in general a multipartite or c-partite tournament is an orientation of a complete c-partite graph. If x is a vertex of a digraph D, then we denote by d+(x) and d−(x) the outdegree and indegree of x, respectively. The global irregularity of a digraph D is defined by ig(D)=max{d+(x),d−(x)}−min{d+(y),d−(y)} over all vertices x and y of D (including x=y). If ig(D)⩽1, then D is called almost regular, and if ig(D)=0, then D is regular.More than 10 years ago, Amar and Manoussakis and independently Wang proved that every arc of a regular bipartite tournament is contained in a directed Hamiltonian cycle. In this paper, we prove that every arc of an almost regular bipartite tournament T is contained in a directed Hamiltonian path if and only if the cardinalities of the partite sets differ by at most one and T is not isomorphic to T3,3, where T3,3 is an almost regular bipartite tournament with three vertices in each partite set.As an application of this theorem and other results, we show that every arc of an almost regular c-partite tournament D with the partite sets V1,V2,…,Vc such that |V1|=|V2|=⋯=|Vc|, is contained in a directed Hamiltonian path if and only if D is not isomorphic to T3,3

Изучение кинетики сорбции отдельных компонентов композитного биосорбента

В данной статье исследуется кинетика сорбции уранил-ионов плесневыми грибами Penicillium pinophilum и Aspergillus niger. Исследования показали что степень сорбции плесневых грибов Aspergillus niger имеет на 3% большую степень сорбции урана, чем Penicillium pinophilum . Так же исследования показали, что после 12 часов сорбция заметно уменьшается и почти останавливается как у одно, так и у другого вида плесневых грибов. This article examines the kinetics of sorption of uranyl ions by fungi Penicillium pinophilum and Aspergillus niger. Studies have shown that the degree of sorption fungi Aspergillus niger has by 3% greater uranium sorption than the Penicillium pinophilum . Studies have shown that after 12 hours of sorption decreases markedly and almost stops as one or the other kind of fungi

Problems of formation of regional budgets

In this work the analysis of regional budgets has been carried out and problems of their formation are defined

On αrγs(k)-perfect graphs

AbstractFor some integer k⩾0 and two graph parameters π and τ, a graph G is called πτ(k)-perfect, if π(H)−τ(H)⩽k for every induced subgraph H of G. For r⩾1 let αr and γr denote the r-(distance)-independence and r-(distance)-domination number, respectively. In (J. Graph Theory 32 (1999) 303–310), I. Zverovich gave an ingenious complete characterization of α1γ1(k)-perfect graphs in terms of forbidden induced subgraphs. In this paper we study αrγs(k)-perfect graphs for r,s⩾1. We prove several properties of minimal αrγs(k)-imperfect graphs. Generalizing Zverovich's main result in (J. Graph Theory 32 (1999) 303–310), we completely characterize α2r−1γr(k)-perfect graphs for r⩾1. Furthermore, we characterize claw-free α2γ2(k)-perfect graphs

Cyclic sums, network sharing and restricted edge cuts in graphs with long cycles

Cyclic Sums, Network Sharing and Restricted Edge Cuts in Graphs with Long Cycles Dieter Rautenbach , Lutz Volkmann Preprint series: 07-06, 8 MSC 2000 05A17 Partitions of integers 05C40 Connectivity Abstract We study graphs G = (V,E) containing a long cycle which for given integers a1, a2, ..., ak 2 N have an edge cut whose removal results in k components with vertex sets V1, V2, ..., Vk such that |Vi| ai for 1 i k. Our results closely relate to problems and recent research in network sharing and network reliability. Keywords: restricted edge connectivity, arbitrarily vertex decomposable graph, network reliability, network sharin

Signed star k-domatic number of a graph

Let $G$ be a simple graph without isolated vertices with vertex set $V(G)$ and edge set $E(G)$ and let $k$ be a positive integer. A function $f:E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star $k$-dominating function on $G$ if $\sum_{e\in E(v)}f(e)\ge k$ for every vertex $v$ of $G$, where $E(v)=\{uv\in E(G)\mid u\in N(v)\}$. A set $\{f_1,f_2,\ldots,f_d\}$ of signed star $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(e)\le 1$ for each $e\in E(G)$, is called a signed star $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed star $k$-dominating family on $G$ is the signed star $k$-domatic number of $G$, denoted by $d_{kSS}(G)$

The influence of insulation of walls of industrial objects on thermal regime at the heating system of gas infrared radiators

The results of a numerical study of the process of heat transfer from the gas infrared emitters in the heated accommodation are represented. Simulation was conducted taking into account the heat withdrawal in the enclosing constructions and of heat exchange with the environment. The estimation of the average values of temperatures of air indoors in the dependence on the different intensity of heat withdrawal into the vertical walls is carried out (when the layer of insulation is present, and without it)