The algorithmic use of hypertree structure and maximum neighbourhood orderings

Brandstädt, Andreas; Chepoi, Victor; Dragan, Feodor F.

doi:10.1007/3-540-59071-4_38

Cited by 39 publications

(57 citation statements)

References 18 publications

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“…In [18], [9] we show that for a given dually chordal graph a maximum neighborhood ordering can be generated in linear time, too.…”

Section: Characterization Of Dually Chordal Graphs Letmentioning

confidence: 99%

“…Using this fact in [9] we present efficient algorithms for r-domination and r-packing problems on dually chordal graphs. Using this fact in [9] we present efficient algorithms for r-domination and r-packing problems on dually chordal graphs.…”

Section: Characterization Of Dually Chordal Graphs Letmentioning

confidence: 99%

“…In the companion papers [18], [19], [9], [10] the algorithmic use of the maximum neighborhood orderings is treated systematically. Many problems remaining NP -complete on chordal graphs have efficient algorithms on strongly chordal graphs.…”

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confidence: 99%

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Dually Chordal Graphs

Brandstädt¹,

Dragan²,

Chepoi³

et al. 1998

SIAM J. Discrete Math.

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120

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Recently in several papers, graphs with maximum neighborhood orderings were characterized and turned out to be algorithmically useful. This paper gives a unified framework for characterizations of those graphs in terms of neighborhood and clique hypergraphs which have the Helly property and whose line graph is chordal. These graphs are dual (in the sense of hypergraphs) to chordal graphs. By using the hypergraph approach in a systematical way new results are obtained, some of the old results are generalized, and some of the proofs are simplified. Introduction.The class of chordal graphs is a by now classical and wellunderstood graph class which is algorithmically useful and has several interesting characterizations. In the theory of relational database schemes there are close relationships between desirable properties of database schemes, acyclicity of corresponding hypergraphs, and chordality of graphs which corresponds to tree and Helly properties of hypergraphs [2], [5], [25]. Chordal graphs arise also in solving large sparse systems of linear equations [28], [36] and in facility location theory [13]. Recently a new class of graphs was introduced and characterized in [20], [6], [21], [39] which is defined by the existence of a maximum neighborhood ordering. These graphs appeared first in [20] and [16] under the name HT -graphs but only a few results have been published in [21]. [34] also introduces maximum neighborhoods but only in connection with chordal graphs (chordal graphs with maximum neighborhood ordering were called there doubly chordal graphs).It is our intention here to attempt to provide a unified framework for characterizations of those graph classes in terms of neighborhood and clique hypergraphs. These graphs are dual (in the sense of hypergraphs) to chordal graphs (this is why we call them dually chordal) but have very different properties-thus they are in general not perfect and not closed under taking induced subgraphs. By using the hypergraph approach in a systematical way new results are obtained, a part of the previous results are generalized, and some of the proofs are simplified. The present paper improves the results of the unpublished manuscripts [20] and [6].Graphs with maximum neighborhood orderings (alias dually chordal graphs) are a generalization of strongly chordal graphs (a well-known subclass of chordal graphs * 2. Standard hypergraph notions and properties. We mainly use the hypergraph terminology of Berge [7]. A finite hypergraph E is a family of nonempty subsets (the edges of E) from some finite underlying set V (the vertices of E). The subhypergraph induced by a set A ⊆ V is the hypergraph E A defined on A by the edge set E A = {e ∩ A : e ∈ E}. The dual hypergraph E * has E as its vertex set and {e ∈ E : v ∈ e} (v ∈ V ) as its edges. The 2-section graph 2SEC(E) of the hypergraph E has vertex set V , and two distinct vertices are adjacent if and only if they are contained in a common edge of E.

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“…In [18], [9] we show that for a given dually chordal graph a maximum neighborhood ordering can be generated in linear time, too.…”

Section: Characterization Of Dually Chordal Graphs Letmentioning

confidence: 99%

Section: Characterization Of Dually Chordal Graphs Letmentioning

confidence: 99%

See 1 more Smart Citation

Dually Chordal Graphs

Brandstädt¹,

Dragan²,

Chepoi³

et al. 1998

SIAM J. Discrete Math.

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120

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“…Suppose that each clique C of C is associated with an integer r(C), where r(C) ≥ 0. Finally, if C consists only of single vertices of G then we obtain the r-domination and r-packing problems [22,19,7,12,11,4] whose particular instances are the domination, k-domination, packing, and k-packing problems [14,9]. …”

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“…First, if C is the family of maximal cliques of G and r(C) = 1 for all C ∈ C then we obtain the clique-transversal and the clique-independence problems [8]. Finally, if C consists only of single vertices of G then we obtain the r-domination and r-packing problems [22,19,7,12,11,4] whose particular instances are the domination, k-domination, packing, and k-packing problems [14,9]. Finally, if C consists only of single vertices of G then we obtain the r-domination and r-packing problems [22,19,7,12,11,4] whose particular instances are the domination, k-domination, packing, and k-packing problems [14,9].…”

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confidence: 99%

Clique r-Domination and Clique r-Packing Problems on Dually Chordal Graphs

Brandstädt¹,

Chepoi²,

Dragan³

1997

SIAM J. Discrete Math.

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Let C be a family of cliques of a graph G = (V, E). Suppose that each clique C of C is associated with an integer r(C), where r(C) ≥ 0. A vertex v r-dominates a clique C of G if d(v, x) ≤ r(C) for all x ∈ C, where d(v, x) is the standard graph distance. A subset D ⊆ V is a clique r-dominating set of G if for every clique C ∈ C there is a vertex u ∈ D which r-dominates C. A clique r-packing set is a subset P ⊆ C such that there are no two distinct cliques C , C ∈ P r-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size and the clique r-packing problem is to find a clique r-packing set with maximum size. The formulated problems include many domination and clique-transversal-related problems as special cases. In this paper an efficient algorithm is proposed for solving these problems on dually chordal graphs which are a natural generalization of strongly chordal graphs. The efficient algorithm is mainly based on the tree structure and special vertex elimination orderings of dually chordal graphs. In some important particular cases where the algorithm works in linear time the obtained results generalize and improve known results on strongly chordal graphs.subset of any other clique. The distance d(u, v) between vertices u, v ∈ V is the length (i.e., number of edges) of a shortest path connecting u and v. The disk centered at vertex v with radius k is the set of all vertices having distance at most k to v:For a clique C and vertex v ∈ V we denote byLet C be a family of cliques of a graph G. Suppose that each clique C of C is associated with an integer r(C), where r(C) ≥ 0 and r(C) > 0 for cliques with size(For cliques of size |C| > 1 and r(C) = 0 there is no vertex v which r-dominates C.) A subset D ⊆ V is a clique r-dominating set of G if for every clique C ∈ C there is a vertex u ∈ D which r-dominates C. A clique r-packing set is a subset P ⊆ C such that there are no two distinct cliques C , C ∈ P r-dominated by a common vertex of G. The clique r-domination problem is to find a clique r-dominating set with minimum size γ C,r (G), and the clique r-packing problem is to find a clique r-packing set with maximum size π C,r (G). Then γ C,r (G) and π C,r (G) are called the clique r-domination and clique r-packing numbers of G.The formulated problems include many domination and clique-transversal-related problems as special cases. First, if C is the family of maximal cliques of G and r(C) = 1 for all C ∈ C then we obtain the clique-transversal and the clique-independence problems [8]. If C is a family of edges of G and r(C) = k (where k is an integer) for all C ∈ C then we obtain the k-neighborhood covering and k-neighborhood independence problems considered in [17]. Finally, if C consists only of single vertices of G then we obtain the r-domination and r-packing problems [22,19,7,12,11,4] whose particular instances are the domination, k-domination, packing, and k-packing problems [14,9]. Dually chordal graphs.In this section we recall the definitions and...

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A story of diameter, radius, and (almost) Helly property

Ducoffe

Dragan

2020

Networks

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We present new algorithmic results for the class of Helly graphs, that is, for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph that makes of the latter an important class of graphs in metric graph theory. We study diameter and radius computations within the Helly graphs, and related graph classes. This is in part motivated by a conjecture on the fine‐grained complexity of these two distance problems within the graph classes of bounded fractional Helly number—that contain as particular cases the proper minor‐closed graph classes and the bounded clique‐width graphs. Note that under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, we first present algorithms which given an n‐vertex m‐edge Helly graph G as input, compute with high probability (w.h.p.) its radius and its diameter in scriptOtrue˜false(mnfalse) time (i.e., subquadratic in n + m). Our algorithms are based on the Helly property and on the unimodality of the eccentricity function in Helly graphs: every vertex of locally minimum eccentricity is a central vertex. Then, we improve our results for the C4‐free Helly graphs, that are exactly the Helly graphs whose balls are convex. For this subclass, we present linear‐time algorithms for computing the eccentricity of all vertices. Doing so, we generalize previous results on strongly chordal graphs to a much larger subclass, that includes, among others, all the bridged Helly graphs and the hereditary Helly graphs. Lastly, we derive approximate versions of our results for the class of chordal graphs: with the latter satisfying an almost‐Helly‐type property, and a stronger (induced‐path) convexity property than the C4‐free Helly graphs. For the chordal graphs, we can compute in quasi linear time the eccentricity of all vertices with an additive one‐sided error of at most one, which is best possible under the strong exponential‐time hypothesis. This answers an open question of Dragan. In fact, we obtain this last result as a byproduct from a more general reduction: from diameter computation on chordal graphs to the Disjoint Sets problem. Roughly, it implies that the split graphs are the only hard instances for diameter computation on chordal graphs. We also get from our reduction that on any subclass of chordal graphs with constant VC‐dimension (and so, for undirected path graphs), the diameter can be computed in truly subquadratic time.

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The algorithmic use of hypertree structure and maximum neighbourhood orderings

Cited by 39 publications

References 18 publications

Dually Chordal Graphs

Dually Chordal Graphs

Clique r-Domination and Clique r-Packing Problems on Dually Chordal Graphs

A story of diameter, radius, and (almost) Helly property

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