The leafage of a chordal graph

Lin, In-Jen; McKee, Terry A.; West, Douglas B.

doi:10.7151/dmgt.1061

Cited by 48 publications

(48 citation statements)

References 23 publications

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“…We will need the following lemma which is folklore (for example, see [12] For more information about clique trees and chordal graphs, see [2,7,13].…”

Section: Definitions and Backgroundmentioning

confidence: 99%

Characterizing directed path graphs by forbidden asteroids

2010

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An asteroidal triple is a stable set of three vertices such that each pair is connected by a path avoiding the neighborhood of the third vertex. Asteroidal triples play a central role in a classical characterization of interval graphs by Lekkerkerker and Boland. Their result says that a chordal graph is an interval graph if and only if it does not contain an asteroidal triple. In this paper, we prove an analogous theorem for directed path graphs which are the intersection graphs of directed paths in a directed tree. For this purpose, we introduce the notion of a special connection. Two non-adjacent vertices are linked by a special connection if either they 103104 JOURNAL OF GRAPH THEORY have a common neighbor or they are the endpoints of two vertex-disjoint chordless paths satisfying certain conditions. A special asteroidal triple is an asteroidal triple such that each pair is linked by a special connection. We prove that a chordal graph is a directed path graph if and only if it does not contain a special asteroidal triple. ᭧

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“…We will need the following lemma which is folklore (for example, see [12] For more information about clique trees and chordal graphs, see [2,7,13].…”

Section: Definitions and Backgroundmentioning

confidence: 99%

Characterizing directed path graphs by forbidden asteroids

2010

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“…Lin, McKee, and West [26] proved that for every chordal graph H with n vertices, the leafage is at most n − lg n − 1 2 lg lg n + O (1). Hence, each chordal graph with n vertices admits a NUF of arity n − lg n − 1 2 lg lg n + O(1).…”

Section: Arity Bounds Based On Leafagementioning

confidence: 99%

“…The leafage l(H) of a chordal graph H is the minimum number of leaves of a tree in which H has an intersection representation; cf. [26].…”

Section: Arity Bounds Based On Leafagementioning

confidence: 99%

Near-Unanimity Functions and Varieties of Reflexive Graphs

Brewster¹,

Feder²,

Hell³

et al. 2008

SIAM J. Discrete Math.

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Abstract. Let H be a graph and k ≥ 3. A near-unanimity function of arity k is a mapping g from the k-tuples over V (H) to V (H) such that g(x 1 , x 2 , . . . , x k ) is adjacent to g(x 1 , x 2 , . . . , x k ) whenever x i x i ∈ E(H) for each i = 1, 2, . . . , k, and g(x 1 , x 2 , . . . , x k ) = a whenever at least k − 1 of the x i 's equal a. Feder and Vardi proved that, if a graph H admits a near-unanimity function, then the homomorphism extension (or retraction) problem for H is polynomial time solvable. We focus on near-unanimity functions on reflexive graphs. The best understood are reflexive chordal graphs H: they always admit a near-unanimity function. We bound the arity of these functions in several ways related to the size of the largest clique and the leafage of H, and we show that these bounds are tight. In particular, it will follow that the arity is bounded by n − √ n + 1, where n = |V (H)|. We investigate substructures forbidden for reflexive graphs that admit a near-unanimity function. It will follow, for instance, that no reflexive cycle of length at least four admits a near-unanimity function of any arity. However, we exhibit nonchordal graphs which do admit near-unanimity functions. Finally, we characterize graphs which admit a conservative near-unanimity function. This characterization has been predicted by the results of Feder, Hell, and Huang. Specifically, those results imply that, if P = NP, the graphs with conservative near-unanimity functions are precisely the so-called bi-arc graphs. We give a proof of this statement without assuming P = NP.

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“…The leafage of a chordal graph G is defined as the smallest integer ℓ such that G has a tree model with ℓ leaves. It was first defined in [19] and it can be computed in polynomial time by the algorithm of [12]. The same algorithm also constructs a tree model of G with minimum number of leaves.…”

Section: Is a Rooted Canonical Solution To G X Of Size Kmentioning

confidence: 99%

Algorithmic complexity of finding cross-cycles in flag complexes

Adamaszek

Stacho

2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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A cross-cycle in a flag simplicial complex K is an induced subcomplex that is isomorphic to the boundary of a cross-polytope and that contains a maximal face of K. A cross-cycle is the most efficient way to define a non-zero class in the homology of K. For an independence complex of a graph G, a cross-cycle is equivalent to an induced matching containing a maximal independent set of G.We study the complexity of finding cross-cycles in independence complexes. We show that in general this problem is NP-complete when input is a graph whose independence complex we consider. This allows us to study special cases. Unfortunately, not a lot has been done in this direction besides the recent polynomial time algorithm for forests [16].In this contribution, we focus on the more general class of chordal graphs. This is a natural choice for the problem as the independence complexes of chordal graphs are quite well understood, namely they are wedges of spheres up to homotopy, and any wedge of spheres can be realized as the independence complex of a chordal graph, up to homotopy.As our main result, we present a polynomial time algorithm for detecting a cross-cycle in the independence complex of a chordal graph. Our algorithm is based on the geometric intersection representation of chordal graphs and has an efficient implementation.We further prove that for chordal graphs cross-cycles detect all of homology of the independence complex. As a corollary, we obtain polynomial time algorithms for such topological properties as contractibility or simple-connectedness of independence complexes of chordal graphs. These problems are undecidable for general independence complexes.We conclude with a discussion of some related cases and open problems.

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The leafage of a chordal graph

Cited by 48 publications

References 23 publications

Characterizing directed path graphs by forbidden asteroids

Characterizing directed path graphs by forbidden asteroids

Near-Unanimity Functions and Varieties of Reflexive Graphs

Algorithmic complexity of finding cross-cycles in flag complexes

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