Graph partitions and integer flows

2004 ◽  
Author(s):  
Xujin Chen
Keyword(s):  
Graph Partitions ◽  
Integer Flows

Integer Flows and Modulo Orientations of Signed Graphs

2021 ◽  
Vol 35 (1) ◽  
pp. 376-391
Author(s):  
Miaomiao Han ◽  
Jiaao Li ◽  
Rong Luo ◽  
Yongtang Shi ◽  
Cun-Quan Zhang
Keyword(s):  
Signed Graphs ◽  
Integer Flows

A heuristic solution procedure for multicommodity integer flows

1995 ◽  
Vol 22 (10) ◽  
pp. 1075-1087 ◽  
Author(s):  
A.K. Aggarwal ◽  
M. Oblak ◽  
R.R. Vemuganti
Keyword(s):  
Solution Procedure ◽  
Heuristic Solution ◽  
Integer Flows

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

2019 ◽  
Vol 29 (02) ◽  
pp. 95-120 ◽  
Author(s):  
Prosenjit Bose ◽  
André van Renssen
Keyword(s):  
Line Segment ◽  
Large Family ◽  
Upper Bounds ◽  
The Other ◽  
Additional Property ◽  
Graph Partitions ◽  
Set Of Points ◽  
The Given

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

scholarly journals Destroying Noncomplete Regular Components in Graph Partitions

2012 ◽  
Vol 72 (2) ◽  
pp. 123-127 ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Graph Partitions

Integer flows

1983 ◽  
Vol 7 (3) ◽  
pp. 349-357 ◽  
Author(s):  
D. H. Younger
Keyword(s):  
Integer Flows

The Price of Order

2016 ◽  
Vol 26 (03n04) ◽  
pp. 135-149
Author(s):  
Prosenjit Bose ◽  
Pat Morin ◽  
André van Renssen
Keyword(s):  
Lower Bounds ◽  
Large Family ◽  
Worst Case ◽  
Graph Partitions ◽  
First Results

We present tight bounds on the spanning ratio of a large family of ordered [Formula: see text]-graphs. A [Formula: see text]-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text]. An ordered [Formula: see text]-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We also show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We provide lower bounds for ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones. For ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered [Formula: see text]-graphs have worse spanning ratios than unordered [Formula: see text]-graphs. Finally, we show that, unlike their unordered counterparts, the ordered [Formula: see text]-graphs with 4, 5, and 6 cones are not spanners.

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