Maximal Irredundant Set Enumeration in Bounded-Degeneracy and Bounded-Degree Hypergraphs

2019 ◽  
pp. 148-159
Author(s):  
Alessio Conte ◽  
Mamadou Moustapha Kanté ◽  
Andrea Marino ◽  
Takeaki Uno
Keyword(s):  
Bounded Degree

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 $\Delta$ .

scholarly journals A short note on conflict‐free coloring on closed neighborhoods of bounded degree graphs

2021 ◽  
Author(s):  
Sriram Bhyravarapu ◽  
Subrahmanyam Kalyanasundaram ◽  
Rogers Mathew
Keyword(s):  
Short Note ◽  
Bounded Degree ◽  
Bounded Degree Graphs

scholarly journals The Complexity of Approximating the Matching Polynomial in the Complex Plane

2021 ◽  
Vol 13 (2) ◽  
pp. 1-37
Author(s):  
Ivona Bezáková ◽  
Andreas Galanis ◽  
Leslie Ann Goldberg ◽  
Daniel Štefankovič
Keyword(s):  
Real Axis ◽  
Complex Plane ◽  
Maximum Degree ◽  
Negative Real Axis ◽  
Positive Real ◽  
Bounded Degree ◽  
Matching Polynomial ◽  
Correlation Decay ◽  
Connective Constant ◽  
General Complex

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in the complex plane. When γ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of γ, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree Δ as long as γ is not a negative real number less than or equal to −1/(4(Δ −1)). Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all Δ ≥ 3 and all real γ less than −1/(4(Δ −1)), the problem of approximating the value of the matching polynomial on graphs of maximum degree Δ with edge parameter γ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real γ, it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of γ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value γ that does not lie on the negative real axis. Our analysis accounts for complex values of γ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

scholarly journals Very fast construction of bounded‐degree spanning graphs via the semi‐random graph process

2020 ◽  
Vol 57 (4) ◽  
pp. 892-919
Author(s):  
Omri Ben‐Eliezer ◽  
Lior Gishboliner ◽  
Dan Hefetz ◽  
Michael Krivelevich
Keyword(s):  
Random Graph ◽  
Bounded Degree

Index-Shuffle Graphs

1997 ◽  
Vol 08 (03) ◽  
pp. 289-304 ◽  
Author(s):  
Marc Baumslag ◽  
Bojana Obrenić
Keyword(s):  
Interconnection Networks ◽  
Graph Embedding ◽  
Theoretical Framework ◽  
Parallel Computers ◽  
Bounded Degree ◽  
Exchange Graph ◽  
Comparative Advantages ◽  
On Line

Index-shuffle graphs are introduced as candidate interconnection networks for parallel computers. The comparative advantages of index-shuffle graphs over the standard bounded-degree "approximations" of the hypercube, namely butterfly-like and shuffle-like graphs, are demonstrated in the theoretical framework of graph embedding and network emulations. An N-node index-shuffle graph emulates: • an N-node shuffle-exchange graph with no slowdown, which the currently best emulations of shuffle-like graphs by hypercubes and butterflies incur a slowdown of Ω( log N). • its like-sized butterfly graph with a slowdown O( log log log N), while the currently best emulations of butterfly-like graphs by shuffle-like graphs incur a slowdown of Ω( log log N). • an N-node hypercube that executes an on-line leveled algorithm with a slowdown O( log log N), while the slowdown of currently best such emulations of the hypercube by its bounded-degree shuffle-like and butterfly-like derivatives remains Ω( log N). Our emulation is based on an embedding of an N-node hypercube into an N-node index-shuffle graph with dilation O( log log N), while the currently best embeddings of the hypercube into its bounded-degree shuffle-like and butterfly-like derivatives incur a dilation of Ω( log N).

Cutwidth II: Algorithms for partial w-trees of bounded degree

2005 ◽  
Vol 56 (1) ◽  
pp. 25-49 ◽  
Author(s):  
Dimitrios M. Thilikos ◽  
Maria Serna ◽  
Hans L. Bodlaender
Keyword(s):  
Bounded Degree
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