Forbidden ordered subgraph vs. forbidden subgraph characterizations of graph classes

1999 ◽  
Vol 30 (2) ◽  
pp. 71-76 ◽  
Author(s):  
Mark Ginn
Keyword(s):  
Forbidden Subgraph ◽  
Graph Classes

scholarly journals A forbidden subgraph characterization of some graph classes using betweenness axioms

2013 ◽  
Vol 313 (8) ◽  
pp. 951-958 ◽  
Author(s):  
Manoj Changat ◽  
Anandavally K. Lakshmikuttyamma ◽  
Joseph Mathews ◽  
Iztok Peterin ◽  
Prasanth G. Narasimha-Shenoi ◽  
...  
Keyword(s):  
Forbidden Subgraph ◽  
Forbidden Subgraph Characterization ◽  
Graph Classes

scholarly journals Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition

10.37236/6431 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Maryam Bahrani ◽  
Jérémie Lumbroso
Keyword(s):  
Asymptotic Analysis ◽  
Forbidden Subgraph ◽  
Decomposition Tree ◽  
Random Generation ◽  
Split Decomposition ◽  
Induced Subgraphs ◽  
Graph Classes ◽  
Cactus Graphs ◽  
Future Work ◽  
Natural Way

Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes.By building on the work of Gioan and Paul (2012) and Chauve et al.(2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph.We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti); in other cases, although the enumerations were known, an abundant potential is unlocked by the grammars we provide (in terms of asymptotic analysis, random generation, and parameter analyses, etc.).We believe this methodology here shows its potential; the natural next step to develop its reach would be to study split-decomposition trees which contain certain prime nodes. This will be the object of future work. 

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 Toughness, Forbidden Subgraphs and Pancyclicity

2021 ◽  
Vol 37 (3) ◽  
pp. 839-866
Author(s):  
Wei Zheng ◽  
Hajo Broersma ◽  
Ligong Wang
Keyword(s):  
Recent Article ◽  
Forbidden Subgraph ◽  
Forbidden Subgraphs ◽  
Free Graph ◽  
Induced Subgraph ◽  
Complete Answer ◽  
Open Case

AbstractMotivated by several conjectures due to Nikoghosyan, in a recent article due to Li et al., the aim was to characterize all possible graphs H such that every 1-tough H-free graph is hamiltonian. The almost complete answer was given there by the conclusion that every proper induced subgraph H of $$K_1\cup P_4$$ K 1 ∪ P 4 can act as a forbidden subgraph to ensure that every 1-tough H-free graph is hamiltonian, and that there is no other forbidden subgraph with this property, except possibly for the graph $$K_1\cup P_4$$ K 1 ∪ P 4 itself. The hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs, as conjectured by Nikoghosyan, was left there as an open case. In this paper, we consider the stronger property of pancyclicity under the same condition. We find that the results are completely analogous to the hamiltonian case: every graph H such that any 1-tough H-free graph is hamiltonian also ensures that every 1-tough H-free graph is pancyclic, except for a few specific classes of graphs. Moreover, there is no other forbidden subgraph having this property. With respect to the open case for hamiltonicity of 1-tough $$K_1\cup P_4$$ K 1 ∪ P 4 -free graphs we give infinite families of graphs that are not pancyclic.

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 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.

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