scholarly journals Algorithmic Aspects of Some Variations of Clique Transversal and Clique Independent Sets on Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 22
Author(s):  
Chuan-Min Lee
Keyword(s):  
Planar Graphs ◽  
Maximum Clique ◽  
Chordal Graphs ◽  
Perfect Graphs ◽  
Line Graphs ◽  
Np Completeness ◽  
Split Graphs ◽  
Transversal Numbers ◽  
The Relationship ◽  
Independence Numbers

This paper studies the maximum-clique independence problem and some variations of the clique transversal problem such as the {k}-clique, maximum-clique, minus clique, signed clique, and k-fold clique transversal problems from algorithmic aspects for k-trees, suns, planar graphs, doubly chordal graphs, clique perfect graphs, total graphs, split graphs, line graphs, and dually chordal graphs. We give equations to compute the {k}-clique, minus clique, signed clique, and k-fold clique transversal numbers for suns, and show that the {k}-clique transversal problem is polynomial-time solvable for graphs whose clique transversal numbers equal their clique independence numbers. We also show the relationship between the signed and generalization clique problems and present NP-completeness results for the considered problems on k-trees with unbounded k, planar graphs, doubly chordal graphs, total graphs, split graphs, line graphs, and dually chordal graphs.

scholarly journals Path (or cycle)-trees with Graph Equations involving Line and Split Graphs

2017 ◽  
Vol 16 (3) ◽  
pp. 1-12
Author(s):  
H P Patil ◽  
V Raja
Keyword(s):  
Line Graphs ◽  
Chromatic Numbers ◽  
Higher Dimensional ◽  
Split Graphs ◽  
Hamiltonian Property ◽  
The Relationship

H-trees generalizes the existing notions of trees, higher dimensional trees and k-ctrees. The characterizations and properties of both Pk-trees for k at least 4 and Cn-trees for n at least 5 and their hamiltonian property, dominations, planarity, chromatic and b-chromatic numbers are established. The conditions under which Pk-trees for k at least 3 (resp. Cn-trees for n at least 4), are the line graphs are determined. The relationship between path-trees and split graphs are developed.

scholarly journals Characterization of Super Strongly Perfect Graphs in Chordal and Strongly Chordal Graphs

2012 ◽  
Vol 11 (4) ◽  
pp. 121-131 ◽  
Author(s):  
R Mary Jeya Jothi ◽  
A Amutha
Keyword(s):  
Dominating Set ◽  
Chordal Graphs ◽  
Perfect Graphs ◽  
Perfect Graph ◽  
Strongly Chordal Graphs ◽  
Minimal Dominating Set ◽  
The Relationship ◽  
Odd Cycles ◽  
Domination Numbers

A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal complete sub graphs of H. In this paper, we have investigated the characterization of Super Strongly Perfect graphs using odd cycles. We have given the characterization of Super Strongly Perfect graphs in chordal and strongly chordal graphs. We have presented the results of Chordal graphs in terms of domination and co - domination numbers γ and . We have given the relationship between diameter, domination and co - domination numbers of chordal graphs. Also we have analysed the structure of Super Strongly Perfect Graph in Chordal graphs and Strongly Chordal graphs.

scholarly journals An efficientg-centroid location algorithm for cographs

2005 ◽  
Vol 2005 (9) ◽  
pp. 1405-1413 ◽  
Author(s):  
V. Prakash
Keyword(s):  
Location Problem ◽  
Maximum Clique ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Outerplanar Graphs ◽  
Arbitrary Graph ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Location Algorithm ◽  
Centroid Location

In 1998, Pandu Rangan et al. Proved that locating theg-centroid for an arbitrary graph is𝒩𝒫-hard by reducing the problem of finding the maximum clique size of a graph to theg-centroid location problem. They have also given an efficient polynomial time algorithm for locating theg-centroid for maximal outerplanar graphs, Ptolemaic graphs, and split graphs. In this paper, we present anO(nm)time algorithm for locating theg-centroid for cographs, wherenis the number of vertices andmis the number of edges of the graph.

Ramsey Numbers for Line Graphs and Perfect Graphs

2012 ◽  
pp. 204-215 ◽  
Author(s):  
Rémy Belmonte ◽  
Pinar Heggernes ◽  
Pim van ’t Hof ◽  
Reza Saei
Keyword(s):  
Perfect Graphs ◽  
Line Graphs ◽  
Ramsey Numbers

scholarly journals Some New Classes of Open Distance-Pattern Uniform Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Bibin K. Jose
Keyword(s):  
Nonempty Subset ◽  
Chordal Graphs ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Induced Subgraphs ◽  
Minimum Cardinality ◽  
Maximal Outerplanar Graphs ◽  
Split Graphs ◽  
Strongly Chordal Graphs

Given an arbitrary nonempty subset M of vertices in a graph G=(V,E), each vertex u in G is associated with the set fMo(u)={d(u,v):v∈M,u≠v} and called its open M-distance-pattern. The graph G is called open distance-pattern uniform (odpu-) graph if there exists a subset M of V(G) such that fMo(u)=fMo(v) for all u,v∈V(G), and M is called an open distance-pattern uniform (odpu-) set of G. The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G). Given some property P, we establish characterization of odpu-graph with property P. In this paper, we characterize odpu-chordal graphs, and thereby characterize interval graphs, split graphs, strongly chordal graphs, maximal outerplanar graphs, and ptolemaic graphs that are odpu-graphs. We also characterize odpu-self-complementary graphs, odpu-distance-hereditary graphs, and odpu-cographs. We prove that the odpu-number of cographs is even and establish that any graph G can be embedded into a self-complementary odpu-graph H, such that G and G¯ are induced subgraphs of H. We also prove that the odpu-number of a maximal outerplanar graph is either 2 or 5.

scholarly journals Algorithms for minimum covering by cliques and maximum clique in claw-free perfect graphs

1981 ◽  
Vol 37 (2-3) ◽  
pp. 181-191 ◽  
Author(s):  
Wen-Lian Hsu ◽  
George L. Nemhauser
Keyword(s):  
Maximum Clique ◽  
Perfect Graphs

Fast and simple algorithms for counting dominating sets in distance-hereditary graphs

2020 ◽  
pp. 2150012
Author(s):  
Min-Sheng Lin
Keyword(s):  
Linear Time ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Dominating Sets ◽  
Complete Problem ◽  
Split Graphs ◽  
Linear Time Algorithms ◽  
Chordal Bipartite Graphs

Counting dominating sets (DSs) in a graph is a #P-complete problem even for chordal bipartite graphs and split graphs, which are both subclasses of weakly chordal graphs. This paper investigates this problem for distance-hereditary graphs, which is another known subclass of weakly chordal graphs. This work develops linear-time algorithms for counting DSs and their two variants, total DSs and connected DSs in distance-hereditary graphs.

scholarly journals Hierarchical complexity of 2-clique-colouring weakly chordal graphs and perfect graphs having cliques of size at least 3

2016 ◽  
Vol 618 ◽  
pp. 122-134
Author(s):  
Hélio B. Macêdo Filho ◽  
Raphael C.S. Machado ◽  
Celina M.H. de Figueiredo
Keyword(s):  
Chordal Graphs ◽  
Perfect Graphs ◽  
Hierarchical Complexity
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