scholarly journals An Introduction to Chordal Graphs and Clique Trees

1993 ◽  
pp. 1-29 ◽  
Author(s):  
Jean R. S. Blair ◽  
Barry Peyton
Keyword(s):  
Chordal Graphs ◽  
Clique Trees

scholarly journals An introduction to chordal graphs and clique trees

1992 ◽  
Author(s):  
J.R.S. Blair ◽  
B.W. Peyton
Keyword(s):  
Chordal Graphs ◽  
Clique Trees

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. 

Clique trees of chordal graphs: leafage and 3-asteroidals

2008 ◽  
Vol 30 ◽  
pp. 237-242 ◽  
Author(s):  
M. Gutierrez ◽  
J.L. Szwarcfiter ◽  
S.B. Tondato
Keyword(s):  
Chordal Graphs ◽  
Clique Trees

scholarly journals An introduction to chordal graphs and clique trees

1992 ◽  
Author(s):  
J.R.S. Blair ◽  
B.W. Peyton
Keyword(s):  
Chordal Graphs ◽  
Clique Trees

Chordal graphs

2021 ◽  
pp. 130-151
Author(s):  
Martin Charles Golumbic
Keyword(s):  
Chordal Graphs

On 3-Degree 4-Chordal Graphs

2020 ◽  
Author(s):  
Devarshi Aggarwal ◽  
R.Mahendra Kumar ◽  
Shwet Prakash ◽  
N. Sadagopan
Keyword(s):  
Chordal Graphs

k-Efficient domination: Algorithmic perspective

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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