A Combinatorial Decomposition Theory

1980 ◽  
Vol 32 (3) ◽  
pp. 734-765 ◽  
Author(s):  
William H. Cunningham ◽  
Jack Edmonds
Keyword(s):  
Undirected Graph ◽  
Decomposition Theory ◽  
Combinatorial Decomposition ◽  
Empty Subset ◽  
Separable Graph

Given a finite undirected graphGandA⊆E(G),G(A)denotes the subgraph ofGhaving edge-setAand having no isolated vertices. For a partition {E1, E2}ofE(G),W(G; E1)denotes the setV(G(E1))⋂V(G(E2)). We say thatGisnon-separableif it is connected and for every proper, non-empty subsetAofE(G), we have |W(G;A)| ≧ 2. Asplitof a non-separable graphGis a partition {E1, E2} ofE(G)such that|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.Where {E1, E2} is a split ofG, W(G; E2)= {u, v}, andeis an element not inE(G),we form graphsGii= 1 and 2, by addingetoG(Ei)as an edge joiningutov.

scholarly journals On the star forest polytope for trees and cycles

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.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

Two Streamlined Depth-First Search Algorithms

1986 ◽  
Vol 9 (1) ◽  
pp. 85-94
Author(s):  
Robert Endre Tarjan
Keyword(s):  
Directed Graph ◽  
Graph Algorithms ◽  
Undirected Graph ◽  
Linear Time ◽  
Search Algorithms ◽  
Depth First Search ◽  
Time Graph

Many linear-time graph algorithms using depth-first search have been invented. We propose simplified versions of two such algorithms, for computing a bipolar orientation or st-numbering of an undirected graph and for finding all feedback vertices of a directed graph.

scholarly journals DETERMINING EQUATIONS FOR HIGHER-ORDER DECOMPOSITIONS OF EXPONENTIAL OPERATORS

1995 ◽  
Vol 09 (25) ◽  
pp. 3241-3268 ◽  
Author(s):  
ZENGO TSUBOI ◽  
MASUO SUZUKI
Keyword(s):  
General Scheme ◽  
Higher Order ◽  
Explicit Formulas ◽  
Decomposition Theory ◽  
Lyndon Words

The general decomposition theory of exponential operators is briefly reviewed. A general scheme to construct independent determining equations for the relevant decomposition parameters is proposed using Lyndon words. Explicit formulas of the coefficients are derived.

On the local solvability of vector fields with critical points

2011 ◽  
Vol 10 (3) ◽  
pp. 785-808
Author(s):  
François Treves
Keyword(s):  
Critical Points ◽  
Vector Fields ◽  
Local Solvability ◽  
Complex Vector ◽  
Base Manifold ◽  
Smooth Complex ◽  
Empty Subset

AbstractThe article discusses the local solvability (or lack thereof) of various classes of smooth, complex vector fields that vanish on some non-empty subset of the base manifold.

MINIMUM CONNECTED r-HOP k-DOMINATING SET IN WIRELESS NETWORKS

2009 ◽  
Vol 01 (01) ◽  
pp. 45-57 ◽  
Author(s):  
DEYING LI ◽  
LIN LIU ◽  
HUIQIANG YANG
Keyword(s):  
Wireless Networks ◽  
Undirected Graph ◽  
Dominating Set ◽  
Approximation Ratio ◽  
Dominating Set Problem ◽  
Simulation Results

In this paper, we study the connected r-hop k-dominating set problem in wireless networks. We propose two algorithms for the problem. We prove that algorithm I for UDG has (2r + 1)3 approximate ratio for k ≤ (2r + 1)2 and (2r + 1)((2r + 1)2 + 1)-approximate ratio for k > (2r + 1)2. And algorithm II for any undirected graph has (2r + 1) ln (Δr) approximation ratio, where Δr is the largest cardinality among all r-hop neighborhoods in the network. The simulation results show that our algorithms are efficient.

scholarly journals Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation

2020 ◽  
Vol 10 (1) ◽  
pp. 311-330 ◽  
Author(s):  
Feng Binhua ◽  
Ruipeng Chen ◽  
Jiayin Liu
Keyword(s):  
Blow Up ◽  
Standing Waves ◽  
Radial Solutions ◽  
Choquard Equation ◽  
Decomposition Theory ◽  
Profile Decomposition ◽  
Strong Instability

Abstract In this paper, we study blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation $$\begin{array}{} \displaystyle i\partial_t\psi- (-{\it\Delta})^s \psi+(I_\alpha \ast |\psi|^{p})|\psi|^{p-2}\psi=0. \end{array}$$ By using localized virial estimates, we firstly establish general blow-up criteria for non-radial solutions in both L2-critical and L2-supercritical cases. Then, we show existence of normalized standing waves by using the profile decomposition theory in Hs. Combining these results, we study the strong instability of normalized standing waves. Our obtained results greatly improve earlier results.

scholarly journals Sharp Bounds on the Diameter of a Graph

1987 ◽  
Vol 30 (1) ◽  
pp. 72-74
Author(s):  
W. F. Smyth
Keyword(s):  
Undirected Graph ◽  
Sharp Bounds ◽  
Image Position

AbstractLet Dn.m, be the diameter of a connected undirected graph on n ≥2 vertices and n - 1 ≤ m ≤ s(n) edges, where s(n) = n(n — l)/2. Then Dn.s(n) = 1, and for ms(n) it is shown thatThe bounds on Dn.m, are sharp.

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