The Maximum Weight Connected Subgraph Problem

The article presents the results of studying the diversity of pear rootstock forms in terms of yield and seed productivity. The research was carried out at the experimental sites of the Orenburg Experimental Station of Horticulture and Viticulture of AllRussian Horticultural Institute for Breeding, Agrotechnology and Nursery and the Botanical Garden of the Orenburg State University in 2017-2019, in typical soil and climatic conditions of the Orenburg city. The purpose of the study is to identify pear rootstock forms characterized by high yield and stable seed productivity for use in the future as a seed rootstock. During the research, 15 pear accessions were studied; the planting scheme was 6x4 m. As a result of research, it was found that the rootstock form Temno-zelenaya is characterized by a high yield (40 kg/tree). High seed productivity of more than 6 seeds in one fruit was observed in samples: Vernaya (6.0-6.5 pcs.), SK-1, SK-3 (6.1-7.8 pcs.), SK-2 (7.0-7.5 pcs.), Chang Bai Li (7.4-7.7 pcs.), Semennaya 214 (7.5-7.8 pcs.). It was revealed that the Xiao he Bai Li variety is characterized by the maximum weight of 1000 seeds (65.2 g). Analysis of accessions by seed yield established that a consistently high yield is observed in the varieties Chang Bai Li (2.5-4.2 %), Vernaya (3.96-4.18 %) and forms SK-1 (2.0-3.25%), SK-2 (2.25-2.75 %), SK-3 (1.43-4.0 %). Pear rootstock forms Chang Bai Li, Vernaya, Semennaya 214, SK-1, SK-2, SK-3 were identifi ed, which can be recommended for production testing as seed pear rootstocks for the conditions of the steppe zone of the Southern Urals.

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On the star forest polytope for trees and cycles

RAIRO - Operations Research ◽

10.1051/ro/2018076 ◽

2019 ◽

Vol 53 (5) ◽

pp. 1763-1773

Author(s):

Meziane Aider ◽

Lamia Aoudia ◽

Mourad Baïou ◽

A. Ridha Mahjoub ◽

Viet Hung Nguyen

Keyword(s):

Undirected Graph ◽

Dominating Set ◽

Discrete Math ◽

Complete Characterization ◽

Facial Structure ◽

Maximum Weight ◽

Single Node ◽

End Node ◽

Vertex Disjoint

Let G = (V, E) be an undirected graph where the edges in E have non-negative weights. A star in G is either a single node of G or a subgraph of G where all the edges share one common end-node. A star forest is a collection of vertex-disjoint stars in G. The weight of a star forest is the sum of the weights of its edges. This paper deals with the problem of finding a Maximum Weight Spanning Star Forest (MWSFP) in G. This problem is NP-hard but can be solved in polynomial time when G is a cactus [Nguyen, Discrete Math. Algorithms App. 7 (2015) 1550018]. In this paper, we present a polyhedral investigation of the MWSFP. More precisely, we study the facial structure of the star forest polytope, denoted by SFP(G), which is the convex hull of the incidence vectors of the star forests of G. First, we prove several basic properties of SFP(G) and propose an integer programming formulation for MWSFP. Then, we give a class of facet-defining inequalities, called M-tree inequalities, for SFP(G). We show that for the case when G is a tree, the M-tree and the nonnegativity inequalities give a complete characterization of SFP(G). Finally, based on the description of the dominating set polytope on cycles given by Bouchakour et al. [Eur. J. Combin. 29 (2008) 652–661], we give a complete linear description of SFP(G) when G is a cycle.

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