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Depletion and/or degradation of natural resources, increasing farm labour scarcity, and high production cost are major threats to the rice-wheat cropping system of north-west India. Conservation agriculture (CA) is a potential solution which can be achieved by switching from puddling then transplanting of rice to dry seeding (DSR), together with changing from conventional tillage (CT) to zero tillage (ZT) for wheat with surface retention of rice residues. Whether the use of ZT for both crops confers additional benefits to either crop is not known. The effects of surface retention of rice residues in wheat on the subsequent DSR crop are also unknown, nor how this is affected by tillage for DSR. Therefore, a field study was conducted during 2012-14 to investigate the interactions between tillage for rice and wheat (CT, ZT), and rice residue management (removed, retained on the soil surface), on the performance of a dry seeded rice-wheat system. There were no significant interaction or main treatment effects on wheat grain yield in the first two years, but in the third year, yield of ZT wheat (5.5 t ha −1 ) was significantly lower than yield of CT wheat (6.0 t ha −1 ). Yield of wheat decreased significantly over the three years, for example from 7.7 to 6.3 to 5.8 t ha −1 in the control system (CT for both crops, no rice straw mulch). Weather data analysis and simulation modelling suggested that this was entirely due to differences in seasonal weather conditions. Growth of non-mulched ZT wheat was inferior to that of mulched ZT wheat, and that of CT wheat with and without mulch, regardless of tillage for rice, although this was not reflected in wheat grain yield. Mulch delayed wheat crop development and the time of irrigation to varying degrees depending on seasonal conditions, and reduced irrigation amount by 50–100 mm in two of the three years. Tillage and rice residue management treatments did not affect the yield of DSR in the first year (mean yield 6.0 t ha −1 ). However, in the second year, growth and grain yield of ZTDSR (3.2 t ha −1 ) were inferior to that of CTDSR (3.8 t ha −1 ), regardless of tillage treatment for wheat. There was a consistent trend for poorer growth of DSR following wheat mulched with rice straw (significant in the first year), suggesting the need to examine the possibility of allelopathic effects of surface rice straw retention in dry seeded rice-wheat systems. There was a decline in rice growth and grain yield over the two years in all treatments. For example, yield of rice in the control declined from 5.9 t ha −1 in 2012 to 4.3 t ha −1 in 2013. Seasonal conditions and simulated growth and yield were similar each year; the poorer observed rice growth and yield in 2013 were due to disease during grain filling, greater weed infestation during the early vegetative stage, and unexplained factors during the middle of the season. Total system rice equivalent yield (REY) was not affected by tillage or mulch treatments nor their interactions in 2012 (mean REY 13.3 t ha −1 ), but was significantly lower in systems with ZTDSR (9.4 t ha −1 ) than CTDSR (10.0 t ha −1 ) in 2013. Treatment effects on irrigation input to rice and the total system were small. The net result was no treatment effects on total cropping system irrigation water productivity (WPi) in 2012 (mean 0.83 kg m −3 ), and significantly higher system WPi in 2013 in systems with CTDSR (0.69 kg m −3 ) than ZTDSR (0.63 kg m −3 ). The results suggest that in a dry seeded rice-wheat cropping system, surface retention of rice residues can improve the growth of ZT wheat right from the first crop, although this did not affect yield in the first three years. However, whether there is an adverse carry over effect of the rice residues on DSR requires further investigation. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G = ( V , E ) is called a 2-point set dominating set of G if for every set T ⊆ V − D there exists a non-empty set S ⊆ D containing at most two vertices such that the induced subgraph 〈 S ∪ T 〉 is connected. A set D ⊆ V ( G ) is called a global 2-point set dominating set of G if D is a 2-point set dominating set of both G and G ‾ . The global 2-point set domination number (2-point set domination number) is the minimum cardinality of a global 2-point set dominating set (2-point set dominating set) in G. In this paper we determine bounds on the global 2-point set domination number of a graph in terms of other graph invariants. We have also given relation between global 2-point set domination number and 2-point set domination number for some classes of graphs. [ABSTRACT FROM AUTHOR]

Given a graph G = ( V , E ) (not necessarily finite), a graphoidal cover of G means a collection Ψ of non-trivial paths in G called Ψ-edges, which are not necessarily open (not necessarily finite), such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. The set of all graphoidal covers of a graph G = ( V , E ) is denoted by G G and for a given Ψ ∈ G G the ordered pair ( G , Ψ) is called a graphoidally covered graph. Two vertices u and v of G are Ψ- adjacent if they are the ends of an open Ψ-edge. A set D of vertices in ( G , Ψ) is Ψ- independent if no two vertices in D are Ψ-adjacent and is said to be Ψ- dominating if every vertex of G is either in D or is Ψ-adjacent to a vertex in D ; G is γ Ψ ( G ) -definable ( γ i Ψ ( G ) -definable) if ( G , Ψ) has a finite Ψ-dominating (Ψ-independent Ψ-dominating) set. Clearly, if G is γ i Ψ ( G ) -definable, then G is γ Ψ ( G ) -definable and γ Ψ ( G ) ≤ γ i Ψ ( G ) . Further if for a graphoidal cover Ψ of G , γ Ψ ( G ) = γ i Ψ ( G ) then we call Ψ as a least-kernel graphoidal cover of G (in short, an LKG cover of G ). A natural question arises: “Does every graph possess an LKG cover?” We firstly exhibit that not every graph possesses an LKG cover and thereafter, in the quest to find graphs possessing an LKG cover, we proved that every finite tree and every finite unicyclic graph admits an LKG cover. We further extend Allan Laskar theorem to infinite graphs by showing every γ ( G ) -definable infinite claw free graph has an LKG cover. [ABSTRACT FROM AUTHOR]

A graphoidal cover of a graph G is a set Ψ of non-trivial paths (which are not necessarily open) in G such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. We denote the set of all graphoidal covers of graph G by G G . In this paper we introduce a parameter gl ( G ), called graphoidal length of the graph G and is defined as g l ( G ) = max Ψ ∈ G G ⁡ { min P ∈ Ψ ⁡ l ( P ) } . We give bounds for the parameter gl ( G ) in terms of the well known and well studied parameter η ( G ) , graphoidal covering number of the graph and show that the bounds are sharp. [ABSTRACT FROM AUTHOR]

In this paper we continue our study of the graph Γ 1 ( R ) defined over a ring R with unity. Γ 1 ( R ) is a simple undirected graph having all the non-zero elements of R as its vertices and two vertices a , b are adjacent if and only if either a b = 0 or b a = 0 or a + b is a unit in R . Concentrating on the graph Γ 1 ( Z n ) , we look at several properties like degrees, girth, Eulerianity, planarity etc. We also look at some properties of the graph Γ 1 ( R ) taken over matrix rings and of the graph Γ 1 ( F ) over a finite field F . [ABSTRACT FROM AUTHOR]