Finding the Hamiltonian circuits in an undirected graph using the mesh-links incidence

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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The genus of graphs associated with vector spaces

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500863 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050086 ◽

Cited By ~ 1

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.

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Two Streamlined Depth-First Search Algorithms

Fundamenta Informaticae ◽

10.3233/fi-1986-9105 ◽

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.

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

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830909000087 ◽

2009 ◽

Vol 01 (01) ◽

pp. 45-57 ◽

Cited By ~ 22

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.

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On Hamiltonian circuits in Cayley diagrams

Discrete Mathematics ◽

10.1016/0012-365x(82)90174-1 ◽

1982 ◽

Vol 38 (1) ◽

pp. 99-108 ◽

Cited By ~ 25

Author(s):

David S. Witte

Keyword(s):

Hamiltonian Circuits

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Sharp Bounds on the Diameter of a Graph

Canadian Mathematical Bulletin ◽

10.4153/cmb-1987-010-0 ◽

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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Consensus Problem of Singular Multi-Agent Systems with Continuous-Time and Fixed Topology

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.278-280.1687 ◽

2013 ◽

Vol 278-280 ◽

pp. 1687-1691

Author(s):

Tong Qiang Jiang ◽

Jia Wei He ◽

Yan Ping Gao

Keyword(s):

Continuous Time ◽

Spanning Tree ◽

Undirected Graph ◽

Necessary Condition ◽

Multi Agent Systems ◽

Agent Systems ◽

Multi Agent ◽

Fixed Topology ◽

The Stability ◽

Sufficient And Necessary Condition

The consensus problems of two situations for singular multi-agent systems with fixed topology are discussed: directed graph without spanning tree and the disconnected undirected graph. A sufficient and necessary condition is obtained by applying the stability theory and the system is reachable asymptotically. But for normal systems, this can’t occur in upper two situations. Finally a simulation example is provided to verify the effectiveness of our theoretical result.

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