scholarly journals A Parameterized Algorithm for Bounded-Degree Vertex Deletion

2016 ◽  
pp. 79-91 ◽  
Author(s):  
Mingyu Xiao
Keyword(s):  
Parameterized Algorithm ◽  
Bounded Degree ◽  
Vertex Deletion

scholarly journals On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

Algorithmica ◽  
2020 ◽  
Author(s):  
Robert Ganian ◽  
Fabian Klute ◽  
Sebastian Ordyniak
Keyword(s):  
Bounded Degree ◽  
Vertex Deletion

scholarly journals On Bounded-Degree Vertex Deletion parameterized by treewidth

2012 ◽  
Vol 160 (1-2) ◽  
pp. 53-60 ◽  
Author(s):  
Nadja Betzler ◽  
Robert Bredereck ◽  
Rolf Niedermeier ◽  
Johannes Uhlmann
Keyword(s):  
Bounded Degree ◽  
Vertex Deletion

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

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