scholarly journals The Firefighter problem on graph classes

2016 ◽  
Vol 613 ◽  
pp. 38-50 ◽  
Author(s):  
Fedor V. Fomin ◽  
Pinar Heggernes ◽  
Erik Jan van Leeuwen
Keyword(s):  
Graph Classes ◽  
Firefighter Problem

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 Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

scholarly journals Algorithms for the rainbow vertex coloring problem on graph classes

2021 ◽  
Author(s):  
Paloma T. Lima ◽  
Erik Jan van Leeuwen ◽  
Marieke van der Wegen
Keyword(s):  
Vertex Coloring ◽  
Coloring Problem ◽  
Graph Classes ◽  
Vertex Coloring Problem

scholarly journals Computing the Grothendieck constant of some graph classes

2011 ◽  
Vol 39 (6) ◽  
pp. 452-456 ◽  
Author(s):  
M. Laurent ◽  
A. Varvitsiotis
Keyword(s):  
Graph Classes

scholarly journals Subcritical Graph Classes Containing All Planar Graphs

2018 ◽  
Vol 27 (5) ◽  
pp. 763-773
Author(s):  
AGELOS GEORGAKOPOULOS ◽  
STEPHAN WAGNER
Keyword(s):  
Planar Graphs ◽  
Graph Classes

We construct minor-closed addable families of graphs that are subcritical and contain all planar graphs. This contradicts (one direction of) a well-known conjecture of Noy.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

scholarly journals Graph classes and the switch Markov chain for matchings

2015 ◽  
Vol 24 (4) ◽  
pp. 885-933 ◽  
Author(s):  
Martin Dyer ◽  
Haiko Müller
Keyword(s):  
Markov Chain ◽  
Graph Classes

scholarly journals Parameterized Complexity of Vertex Deletion into Perfect Graph Classes

2011 ◽  
pp. 240-251 ◽  
Author(s):  
Pinar Heggernes ◽  
Pim van’t Hof ◽  
Bart M. P. Jansen ◽  
Stefan Kratsch ◽  
Yngve Villanger
Keyword(s):  
Parameterized Complexity ◽  
Perfect Graph ◽  
Graph Classes ◽  
Vertex Deletion

scholarly journals On edge-intersection graphs of k-bend paths in grids

2010 ◽  
Vol Vol. 12 no. 1 ◽  
Author(s):  
Therese Biedl ◽  
Michal Stern
Keyword(s):  
Conflict Resolution ◽  
Planar Graph ◽  
Bipartite Graph ◽  
Outerplanar Graph ◽  
Line Graphs ◽  
Intersection Graphs ◽  
Grid Networks ◽  
Graph Classes ◽  
International Audience ◽  
Number Of Bends

International audience Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks. In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k. We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with -regular edge orientations.

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