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The induced path number $$\rho (G)$$ of a graph G is defined as the minimum number of subsets into which the vertex set of G can be partitioned so that each subset induces a path. A product Nordhaus-Gaddum-type result is a bound on the product of a parameter of a graph and its complement. Hattingh et al. (Util Math 94:275-285, ) showed that if G is a graph of order n, then $$\lceil \frac{n}{4} \rceil \le \rho (G) \rho (\overline{G}) \le n \lceil \frac{n}{2} \rceil $$ , where these bounds are best possible. It was also noted that the upper bound is achieved when either G or $$\overline{G}$$ is a graph consisting of n isolated vertices. In this paper, we determine best possible upper and lower bounds for $$\rho (G) \rho (\overline{G})$$ when either both G and $$\overline{G}$$ are connected or neither G nor $$\overline{G}$$ has isolated vertices. [ABSTRACT FROM AUTHOR]

Let G = ( V, E) be a graph. A set $${S \subseteq V}$$ is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ tr( G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ tr( G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ tr( G) ≤ n − diam( G) − r + 1. [ABSTRACT FROM AUTHOR]

This study compares the effects of ethylenediamine tetra-acetate (EDTA) and heparin in various concentrations as anticoagulants on the microhaematocrit values of the blood of five species of freshwater fish. EDTA always produced haemolysis after a time interval ranging from 1 to 48 hrs and increased cell volume. Heparin in powdered form was found to be the anticoagulant of choice although minor haematological changes do occur, but it is stressed that preliminary experiments must be done to examine the effects of the anticoagulant before routine haematologic analyses are preformed. [ABSTRACT FROM AUTHOR]

The effectiveness of EDTA (dipotassium and disodium salts) and heparin as anticoagulants for the blood of puffadders and pigeons has been investigated. In the case of the snakes all three substances produced haemolysis after variable time intervals, and increases in cell volume were observed. In the case of the pigcons, samples stored at 4° C showed lower haematocrit values and haemolysed sooner than samples stored at room temperature. [ABSTRACT FROM AUTHOR]