Recognition of $C_4$-Free and 1/2-Hyperbolic Graphs

Coudert, David; Ducoffe, Guillaume

doi:10.1137/140954787

Cited by 27 publications

(28 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, for every 4-tuple u, x, v, y of G. Coudert and Ducoffe [CD14] show that if one can detect an induced C 4 in O(n 3−ε ) time for some ε > 0, then one can also determine whether a graph is 1/2-hyperbolic in O(n 3−δ ) time for some δ > 0. Together with this reduction, our results immediately imply the first subcubic algorithm for 1/2-hyperbolicity.…”

Section: Introductionmentioning

confidence: 99%

Finding Four-Node Subgraphs in Triangle Time

Williams¹,

Wang²,

Williams³

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

We present new algorithms for finding induced four-node subgraphs in a given graph, which run in time roughly that of detecting a clique on three nodes (i.e., a triangle).• The best known algorithms for triangle finding in an nnode graph take O(n ω ) time, where ω < 2.373 is the matrix multiplication exponent. We give a general randomized technique for finding any induced four-node subgraph, except for the clique or independent set on 4 nodes, iñ O(n ω ) time with high probability. The algorithm can be derandomized in some cases: we show how to detect a diamond (or its complement) in deterministicÕ(n ω ) time.Our approach substantially improves on prior work. For instance, the previous best algorithm for C 4 detection ran in O(n 3.3 ) time, and for diamond detection in O(n 3 ) time.• For sparse graphs with m edges, the best known triangle finding algorithm runs in O(m 2ω/(ω+1) ) ≤ O(m 1.41 ) time. We give a randomizedÕ(m 2ω/(ω+1) ) time algorithm (analogous to the best known for triangle finding) for finding any induced four-node subgraph other than C 4 , K 4 and their complements. In the case of diamond detection, we also design a deterministicÕ(m 2ω/(ω+1) ) time algorithm. For C 4 or its complement, we give randomized O(m (4ω−1)/(2ω+1) ) ≤ O(m 1.48 ) time finding algorithms. These algorithms substantially improve on prior work. For instance, the best algorithm for diamond detection ran in O(m 1.5 ) time.

show abstract

Section: Introductionmentioning

confidence: 99%

Finding Four-Node Subgraphs in Triangle Time

Williams¹,

Wang²,

Williams³

et al. 2014

Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

show abstract

“…Currently, the best-known theoretical algorithm for the problem runs in O(n 3.69 )-time [23], and the best-known practical algorithm has O(n 4 )-time complexity [14]. Recognizing graphs with small hyperbolicity upper-bounded by 1 2 is computationally equivalent to decide whether there is a chordless cycle of length 4 in a graph, and it can be done in O(n 3.26 )-time by using fast rectangular matrix multiplication [15,33]. Note also that the hyperbolicity of a connected graph is the maximum hyperbolicity taken over all its biconnected components.…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…This both comes from its challenging implementation, relying on fast square matrix multiplications, and its time complexity which is strictly more than cubic. While practical advances have been made, improving the computational cost on certain graph classes (see [14,31]), a recent theoretical work [15] suggests that an algorithm for the problem with a significant speed-up for all graphs is unlikely to exist. This motivates the study of structural properties that may help to decrease the running time of the computation of hyperbolicity.…”

Section: Introductionmentioning

confidence: 99%

Applying clique-decomposition for computing Gromov hyperbolicity

Cohen

Coudert

Ducoffe

et al. 2017

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

The shortest-path metric d of a connected graph G is δ-hyperbolic if, and only if, it satisfies d (u, v)+d(x, y) ≤ max{d(u, x)+d(v, y), d(u, y)+d(v, x)}+2δ, for every 4-tuple u, x, v, y of G. We investigate some relations between the hyperbolicity of a graph and the hyperbolicity of its atoms, that are the subgraphs resulting from the clique-decomposition invented by Tarjan [34,45]. More precisely, we prove that the maximum hyperbolicity taken over all the atoms is at least the hyperbolicity of G minus one. We also give an algorithm to slightly modify the atoms, which is at no extra cost than computing the atoms themselves, and so that the maximum hyperbolicity taken over all the resulting graphs is exactly the hyperbolicity of G. An experimental evaluation of our methodology is provided for large collaboration networks. Finally, we deduce from our theoretical results the first linear-time algorithm to compute the hyperbolicity of an outerplanar graph. , pour tout quadruplet u, x, v, y de G. Nous étudions la relation entre l'hyperbolicité d'un graphe et celle de chacun de ses atomes. Ces derniers sont les sous-graphes résultant de la décompo-sition d'un graphe par des cliques-séparatrices [34,45]. Plus précisemment, nous montrons que l'hyperbolicité d'un atome est au plus l'hyperbolicité de G moins un. Nous proposons un algorithme pour modifier les atomes de sorte que la valeur maximale de l'hyperbolicité de ces atomes modifiés soit exactement l'hyperbolicité de G. La complexité de cet algorithme est la même que celle de la décomposition du graphe par des cliques-séparatrices. Nous évaluons expériementale-ment cette méthode sur des graphes de collaborations (co-auteurs). Enfin, nous proposons un algorithme pour calculer en temps linéaire l'hyperbolicité des graphes planaires extérieurs.

show abstract

“…In [48] it was proved the equivalence of the hyperbolicity of many negatively curved surfaces and the hyperbolicity of a graph related to it; hence, it is useful to know hyperbolicity criteria for graphs from a geometrical viewpoint. In recent years, the study of mathematical properties of Gromov hyperbolic spaces has become a topic of increasing interest in graph theory and its applications (see, e.g., [3,7,9,15,16,17,18,20,21,26,28,30,32,42,43,44,47,48,50] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

“…For a finite graph with n vertices it is possible to compute δ(G) in time O(n 3.69 ) [20] (this is improved in [16,17]). Given a Cayley graph (of a presentation with solvable word problem) there is an algorithm which allows to decide if it is hyperbolic [39].…”

Section: Introductionmentioning

confidence: 99%

On the hyperbolicity constant of circular-arc graphs

Reyes

Rodríguez

Sigarreta

et al. 2019

Discrete Applied Mathematics

View full text Add to dashboard Cite

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. It measures the tree-likeness of a graph from a metric viewpoint. In particular, we are interested in circular-arc graphs, which is an important class of geometric intersection graphs. In this paper we give sharp bounds for the hyperbolicity constant of (finite and infinite) circular-arc graphs. Moreover, we obtain bounds for the hyperbolicity constant of the complement and line of any circular-arc graph. In order to do that, we obtain new results about regular, chordal and line graphs which are interesting by themselves.

show abstract

Recognition of $C_4$-Free and 1/2-Hyperbolic Graphs

Cited by 27 publications

References 35 publications

Finding Four-Node Subgraphs in Triangle Time

Finding Four-Node Subgraphs in Triangle Time

Applying clique-decomposition for computing Gromov hyperbolicity

On the hyperbolicity constant of circular-arc graphs

Contact Info

Product

Resources

About