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A note of arbitrarily vertex decomposable graphs

Marczyk A.

Opuscula Mathematica

2006

Vol. 26, no. 1

109-118

A graph G of order n is said to be arbitrarily vertex decomposable if for each sequence (n1,..., nk) of positive integers such that n1 + ... + nk = n there exists a partition (V1,..., Vk) of the vertex set of G such that for each i ∈ {1,..., k}, Vi induces a connected subgraph of G on ni vertices. In this paper we show that if G is a two-connected graph on n vertices with the independence number at most ⌈n/2⌉ and such that the degree sum of any pair of non-adjacent vertices is at least n - 3, then G is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition, where the bound n - 3 is replaced by n - 2.

Przemiany związków azotowych w osadzie czynnym w warunkach napowietrzania

Marczyk A., Bernat K.

Inżynieria i Aparatura Chemiczna

2005

Nr 4s

54-55

The efficiency of nitrogen compounds removal by activated sludge of municipal sewage at stable amount of air supplied to the sequential batch reactor was investigated. It was proved that nitrification and denitrification took place in activated sludge during aeration. Nitrogen compounds were introduced into municipal sewage as ammonium, nitrate and nitrite forms. The efficiency of nitrogen compounds removal in tested technological conditions depended on the form of nitrogen.