scholarly journals An Optimal O ( nm ) Algorithm for Enumerating All Walks Common to All Closed Edge-covering Walks of a Graph

2019 ◽  
Vol 15 (4) ◽  
pp. 1-17 ◽  
Author(s):  
Massimo Cairo ◽  
Paul Medvedev ◽  
Nidia Obscura Acosta ◽  
Romeo Rizzi ◽  
Alexandru I. Tomescu
Keyword(s):  
Edge Covering

scholarly journals Edge covering pseudo-outerplanar graphs with forests

2012 ◽  
Vol 312 (18) ◽  
pp. 2788-2799 ◽  
Author(s):  
Xin Zhang ◽  
Guizhen Liu ◽  
Jian-Liang Wu
Keyword(s):  
Outerplanar Graphs ◽  
Edge Covering

Finding a Covering Triangulation Whose Maximum Angle is Provably Small

1997 ◽  
Vol 07 (01n02) ◽  
pp. 5-20 ◽  
Author(s):  
Scott A. Mitchell
Keyword(s):  
Line Graph ◽  
Local Geometry ◽  
Common Boundary ◽  
Straight Line ◽  
Maximum Angle ◽  
Multiple Regions ◽  
Explicit Lower Bound ◽  
Vertex Set ◽  
Edge Covering ◽  
Planar Straight Line Graph

We consider the following problem: given a planar straight-line graph, find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem differs from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound γopt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle γ is provably close to γopt. Specifically, we show that [Formula: see text], i.e., our γ is not much closer to π than is γopt. To our knowledge, this result represents the first nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log n), where n is the number of vertices of the input. We have implemented an O(n2) time version of our algorithm.

scholarly journals A distributional study of the path edge-covering numbers for random trees

2008 ◽  
Vol 156 (7) ◽  
pp. 1036-1052
Author(s):  
Alois Panholzer
Keyword(s):  
Random Trees ◽  
Covering Numbers ◽  
Edge Covering

A dynamic edge covering and scheduling problem: complexity results and approximation algorithms

2013 ◽  
Vol 8 (4) ◽  
pp. 1201-1212
Author(s):  
Jiaming Qiu ◽  
Thomas C. Sharkey
Keyword(s):  
Approximation Algorithms ◽  
Scheduling Problem ◽  
Problem Complexity ◽  
Complexity Results ◽  
Edge Covering

scholarly journals The Minimum Edge Covering Energy of a Graph

2021 ◽  
Vol 45 (6) ◽  
pp. 969-975
Author(s):  
SAMIRA SABETI ◽  
◽  
AKRAM BANIHASHEMI DEHKORDI ◽  
SAEED MOHAMMADIAN SEMNANI
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Underlying Graph ◽  
Graph Energy ◽  
Edge Covering

In this paper, we introduce a new kind of graph energy, the minimum edge covering energy, ECE(G). It depends both on the underlying graph G, and on its particular minimum edge covering CE. Upper and lower bounds for ECE(G) are established. The minimum edge covering energy and some of the coefficients of the polynomial of well-known families of graphs like Star, Path and Cycle Graphs are computed

Edge covering problem under hybrid uncertain environments

2013 ◽  
Vol 219 (11) ◽  
pp. 6044-6052 ◽  
Author(s):  
Yaodong Ni
Keyword(s):  
Covering Problem ◽  
Uncertain Environments ◽  
Edge Covering

scholarly journals GENERALIZED WATCHMAN ROUTE PROBLEM WITH DISCRETE VIEW COST

2010 ◽  
Vol 20 (02) ◽  
pp. 119-146 ◽  
Author(s):  
PENGPENG WANG ◽  
RAMESH KRISHNAMURTI ◽  
KAMAL GUPTA
Keyword(s):  
Travel Cost ◽  
Approximation Ratio ◽  
Planning Problem ◽  
Weighted Sum ◽  
Np Hard ◽  
View Planning ◽  
Total Cost ◽  
Total Length ◽  
Route Problem ◽  
Edge Covering

In this paper, we introduce a generalized version of the Watchman Route Problem (WRP) where the objective is to plan a continuous closed route in a polygon (possibly with holes) and a set of discrete viewpoints on the planned route such that every point on the polygon boundary is visible from at least one viewpoint. Each planned viewpoint has some associated cost. The total cost to minimize is a weighted sum of the view cost, proportional to the number of viewpoints, and the travel cost, the total length of the route. We call this problem the Generalized Watchman Route Problem or the GWRP. We tackle a restricted nontrivial (it remains NP-hard and log-inapproximable) version of GWRP where each polygon edge is entirely visible from at least one planned viewpoint. We call it Whole Edge Covering GWRP. The algorithm we propose first constructs a graph that connects O(n12) number of sample viewpoints in the polygon, where n is the number of polygon vertices; and then solves the corresponding View Planning Problem with Combined View and Traveling Cost, using an LP-relaxation based algorithm we introduced in [27, 29]. We show that our algorithm has an approximation ratio in the order of either the view frequency, defined as the maximum number of sample viewpoints that cover a polygon edge, or a polynomial of log n, whichever is smaller.

EDGE COVERING OF COMPLEX TRIANGLES IN RECTANGULAR DUAL FLOORPLANNING

1993 ◽  
Vol 03 (03) ◽  
pp. 721-731 ◽  
Author(s):  
YACHYANG SUN ◽  
KOK-HOO YEAP
Keyword(s):  
Dual Graph ◽  
Input Graph ◽  
Adjacency Graph ◽  
Edge Covering ◽  
Np Complete ◽  
Face Complex ◽  
Elimination Problem

Rectangular dual graph approach to floorplanning is based on the adjacency graph of the modules in a floorplan. If the input adjacency graph contains a cycle of length three which is not a face (complex triangle), a rectangular floorplan does not exist. Thus, complex triangles have to be eliminated before applying any floorplanning algorithm. This paper shows that the weighted complex triangle elimination problem is NP-complete, even when the input graphs are restricted to 1-level containment. For adjacency graph with 0-level containment, the unweighted problem is optimally solvable in O(c1.5 + n) time where c is the number of complex triangles and n is the number of vertices of the input graph.

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