Neighborhood inclusion posets and tree representations for chordal and dually chordal graphs

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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Chordal graphs

Topics in Algorithmic Graph Theory ◽

10.1017/9781108592376.009 ◽

2021 ◽

pp. 130-151

Author(s):

Martin Charles Golumbic

Keyword(s):

Chordal Graphs

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On 3-Degree 4-Chordal Graphs

2020 IEEE 4th Conference on Information & Communication Technology (CICT) ◽

10.1109/cict51604.2020.9312070 ◽

2020 ◽

Author(s):

Devarshi Aggarwal ◽

R.Mahendra Kumar ◽

Shwet Prakash ◽

N. Sadagopan

Keyword(s):

Chordal Graphs

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Fully dynamic algorithm for chordal graphs with O(1) query-time and O(n2) update-time

Theoretical Computer Science ◽

10.1016/j.tcs.2012.05.002 ◽

2012 ◽

Vol 445 ◽

pp. 82-92 ◽

Cited By ~ 2

Author(s):

Mauro Mezzini

Keyword(s):

Chordal Graphs ◽

Query Time ◽

Dynamic Algorithm

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k-Efficient domination: Algorithmic perspective

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500513 ◽

2022 ◽

Author(s):

Mohsen Alambardar Meybodi

Keyword(s):

Decision Problem ◽

Dominating Set ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Chordal Graphs ◽

Sparse Graphs ◽

Fpt Algorithm ◽

Domination Problem ◽

Efficient Dominating Set ◽

Np Complete

A set [Formula: see text] of a graph [Formula: see text] is called an efficient dominating set of [Formula: see text] if every vertex [Formula: see text] has exactly one neighbor in [Formula: see text], in other words, the vertex set [Formula: see text] is partitioned to some circles with radius one such that the vertices in [Formula: see text] are the centers of partitions. A generalization of this concept, introduced by Chellali et al. [k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122], is called [Formula: see text]-efficient dominating set that briefly partitions the vertices of graph with different radiuses. It leads to a partition set [Formula: see text] such that each [Formula: see text] consists a center vertex [Formula: see text] and all the vertices in distance [Formula: see text], where [Formula: see text]. In other words, there exist the dominators with various dominating powers. The problem of finding minimum set [Formula: see text] is called the minimum [Formula: see text]-efficient domination problem. Given a positive integer [Formula: see text] and a graph [Formula: see text], the [Formula: see text]-efficient Domination Decision problem is to decide whether [Formula: see text] has a [Formula: see text]-efficient dominating set of cardinality at most [Formula: see text]. The [Formula: see text]-efficient Domination Decision problem is known to be NP-complete even for bipartite graphs [M. Chellali, T. W. Haynes and S. Hedetniemi, k-Efficient partitions of graphs, Commun. Comb. Optim. 4 (2019) 109–122]. Clearly, every graph has a [Formula: see text]-efficient dominating set but it is not correct for efficient dominating set. In this paper, we study the following: [Formula: see text]-efficient domination problem set is NP-complete even in chordal graphs. A polynomial-time algorithm for [Formula: see text]-efficient domination in trees. [Formula: see text]-efficient domination on sparse graphs from the parametrized complexity perspective. In particular, we show that it is [Formula: see text]-hard on d-degenerate graphs while the original dominating set has Fixed Parameter Tractable (FPT) algorithm on d-degenerate graphs. [Formula: see text]-efficient domination on nowhere-dense graphs is FPT.

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An Algorithm for the Enumeration of Serial and/or Parallel Combinations of Kinematic Building Blocks

Volume 2: 28th Biennial Mechanisms and Robotics Conference, Parts A and B ◽

10.1115/detc2004-57295 ◽

2004 ◽

Author(s):

Hong-Sen Yan ◽

Feng-Ming Ou ◽

Ming-Feng Tang

Keyword(s):

Graph Theory ◽

Building Blocks ◽

Power Source ◽

Cam Follower ◽

Tree Representations ◽

The Given

An algorithm is presented, based on graph theory, for enumerating all feasible serial and/or parallel combined mechanisms from the given rotary or translational power source and specific kinematic building blocks. Through the labeled out-tree representations for the configurations of combined mechanisms, the enumeration procedure is developed by adapting the algorithm for the enumeration of trees. A rotary power source and four kinematic building blocks: a crank-rocker linkage, a rack-pinion, a double-slider mechanism, and a cam-follower mechanism, are chosen as the combination to illustrate the algorithm. And, two examples are provided to validate the algorithm.

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