scholarly journals A Note on Independence Complexes of Chordal Graphs and Dismantling

10.37236/5571 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Michał Adamaszek
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Homotopy Equivalent ◽  
Tree Models ◽  
Independence Complex

We show that the independence complex of a chordal graph is contractible if and only if this complex is dismantlable (strong collapsible) and it is homotopy equivalent to a sphere if and only if its core is a cross-polytopal sphere. The proof uses the properties of tree models of chordal graphs.

Clique Partitions of Chordal Graphs

1993 ◽  
Vol 2 (4) ◽  
pp. 409-415 ◽  
Author(s):  
Paul Erdős ◽  
Edward T. Ordman ◽  
Yechezkel Zalcstein
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Split Graph ◽  
Threshold Graph

To partition the edges of a chordal graph on n vertices into cliques may require as many as n2/6 cliques; there is an example requiring this many, which is also a threshold graph and a split graph. It is unknown whether this many cliques will always suffice. We are able to show that (1 − c)n2/4 cliques will suffice for some c > 0.

Bonferroni-Type Inequalities via Chordal Graphs

2002 ◽  
Vol 11 (4) ◽  
pp. 349-351 ◽  
Author(s):  
KLAUS DOHMEN
Keyword(s):  
Type Inequality ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Finite Collection ◽  
Selection Of

Let {Av}v∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

scholarly journals A $\beta$ Invariant for Greedoids and Antimatroids

10.37236/1298 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Gary Gordon
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Combinatorial Interpretations

We extend Crapo's $\beta $ invariant from matroids to greedoids, concentrating especially on antimatroids. Several familiar expansions for $\beta (G)$ have greedoid analogs. We give combinatorial interpretations for $\beta (G)$ for simplicial shelling antimatroids associated with chordal graphs. When $G$ is this antimatroid and $b(G)$ is the number of blocks of the chordal graph $G$, we prove $\beta (G)=1-b(G)$.

scholarly journals An Edge-Signed Generalization of Chordal Graphs, Free Multiplicities on Braid Arrangements, and Their Characterizations

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Takuro Abe ◽  
Koji Nuida ◽  
Yasuhide Numata
Keyword(s):  
Special Kind ◽  
Chordal Graph ◽  
Hyperplane Arrangements ◽  
Chordal Graphs ◽  
Nous Proposons ◽  
Signed Graphs ◽  
International Audience ◽  
Perfect Elimination Ordering

International audience In this article, we propose a generalization of the notion of chordal graphs to signed graphs, which is based on the existence of a perfect elimination ordering for a chordal graph. We give a special kind of filtrations of the generalized chordal graphs, and show a characterization of those graphs. Moreover, we also describe a relation between signed graphs and a certain class of multiarrangements of hyperplanes, and show a characterization of free multiarrangements in that class in terms of the generalized chordal graphs, which generalizes a well-known result by Stanley on free hyperplane arrangements. Finally, we give a remark on a relation of our results with a recent conjecture by Athanasiadis on freeness characterization for another class of hyperplane arrangements. Dans cet article, nous proposons une généralisation de la notion des graphes triangulés à graphes signés, qui est basée sur l'existence d'un ordre d'élimination simplicial à un graphe triangulé. Nous donnons un genre spécial de filtrations des graphes triangulés généralisés, et montrons une caractérisation de ces graphes. De plus, nous décrivons aussi une relation entre graphes signés et une certaine classe de multicompositions d'hyperplans, et montrons une caractérisation de multicompositions libres dans cette classe en termes des graphes triangulés généralisés, qui généralise un résultat célèbre de Stanley sur compositions libres d'hyperplans. Finalement, nous donnons une remarque sur une relation de nos résultats avec une conjecture récente d'Athanasiadis sur une caractérisation du freeness d'une autre classe de compositions d'hyperplans.

scholarly journals Clique Trees of Infinite Locally Finite Chordal Graphs

10.37236/3928 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Christoph Hofer-Temmel ◽  
Florian Lehner
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Locally Finite ◽  
Clique Trees ◽  
Clique Tree

We investigate clique trees of infinite locally finite chordal graphs. Our main contribution is a bijection between the set of clique trees and the product of local finite families of finite trees. Even more, the edges of a clique tree are in bijection with the edges of the corresponding collection of finite trees. This allows us to enumerate the clique trees of a chordal graph and extend various classic characterisations of clique trees to the infinite setting. 

scholarly journals Collapsibility of Non-Cover Complexes of Graphs

10.37236/8684 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Ilkyoo Choi ◽  
Jinha Kim ◽  
Boram Park
Keyword(s):  
Simplicial Complex ◽  
Domination Number ◽  
Independent Set ◽  
Chordal Graphs ◽  
Alexander Dual ◽  
Vertex Set ◽  
Independence Complex ◽  
Independence Domination Number

Let $G$ be a graph on the vertex set $V$. A vertex subset $W \subseteq V$ is a cover of $G$ if $V \setminus W$ is an independent set of $G$, and $W$ is a non-cover of $G$ if $W$ is not a cover of $G$. The non-cover complex of $G$ is a simplicial complex on $V$ whose faces are non-covers of $G$. Then the non-cover complex of $G$ is the combinatorial Alexander dual of the independence complex of $G$. Aharoni asked if the non-cover complex of a graph $G$ without isolated vertices is $(|V(G)|-i\gamma(G)-1)$-collapsible where $i\gamma(G)$ denotes the independence domination number of $G$. Extending a result by the second author, who verified Aharoni's question in the affirmative for chordal graphs, we prove that the answer to the question is yes for all graphs.

scholarly journals Chordal and Sequentially Cohen-Macaulay Clutters

10.37236/695 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Russ Woodroofe
Keyword(s):  
Chordal Graphs ◽  
Geometric Properties ◽  
Linear Resolution ◽  
Independence Complex ◽  
Definition Of

We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially Cohen-Macaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal non-chordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices.

scholarly journals An Upper Bound for the Regularity of Symbolic Powers of Edge Ideals of Chordal Graphs

10.37236/8566 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Seyed Amin Seyed Fakhari
Keyword(s):  
Upper Bound ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Symbolic Power ◽  
Matching Number ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Symbolic Powers ◽  
Reg I

Assume that $G$ is a chordal graph with edge ideal $I(G)$ and ordered matching number $\nu_{o}(G)$. For every integer $s\geq 1$, we denote the $s$-th symbolic power of $I(G)$ by $I(G)^{(s)}$. It is shown that ${\rm reg}(I(G)^{(s)})\leq 2s+\nu_{o}(G)-1$. As a consequence, we determine the regularity of symbolic powers of edge ideals of chordal Cameron-Walker graphs.

scholarly journals On the hyperbolicity of edge-chordal and path-chordal graphs

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2599-2607 ◽  
Author(s):  
Sergio Bermudo ◽  
Walter Carballosa ◽  
José Rodríguez ◽  
José Sigarreta
Keyword(s):  
Chordal Graph ◽  
Chordal Graphs ◽  
Gromov Hyperbolicity ◽  
Geodesic Triangle ◽  
Geodesic Metric Space ◽  
Geodesic Metric ◽  
Hyperbolic Graph ◽  
Hyperbolic Graphs ◽  
Definition Of ◽  
Two Sides

If X is a geodesic metric space and x1, x2, x3 ( X, a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is ?-hyperbolic (in the Gromov sense) if any side of T is contained in a ?-neighborhood of the union of the other two sides, for every geodesic triangle T in X. An important problem in the study of hyperbolic graphs is to relate the hyperbolicity with some classical properties in graph theory. In this paper we find a very close connection between hyperbolicity and chordality: we extend the classical definition of chordality in two ways, edge-chordality and path-chordality, in order to relate this propertywith Gromov hyperbolicity. In fact, we prove that every edge-chordal graph is hyperbolic and that every hyperbolic graph is path-chordal. Furthermore, we prove that every path-chordal cubic graph with small path-chordality constant is hyperbolic.

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