Generation of maximal fuzzy cliques of fuzzy permutation graph and applications

2019 ◽  
Vol 53 (3) ◽  
pp. 1585-1614 ◽  
Author(s):  
Sreenanda Raut ◽  
Madhumangal Pal
Keyword(s):  
Permutation Graph

scholarly journals Complexity of Hamiltonian Cycle Reconfiguration

Algorithms ◽  
2018 ◽  
Vol 11 (9) ◽  
pp. 140 ◽  
Author(s):  
Asahi Takaoka
Keyword(s):  
Hamiltonian Cycle ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Interval Graph ◽  
Unit Interval ◽  
Chordal Graphs ◽  
Hamiltonian Cycles ◽  
Permutation Graph ◽  
Proper Subclass ◽  
Chordal Bipartite Graphs

The Hamiltonian cycle reconfiguration problem asks, given two Hamiltonian cycles C 0 and C t of a graph G, whether there is a sequence of Hamiltonian cycles C 0 , C 1 , … , C t such that C i can be obtained from C i − 1 by a switch for each i with 1 ≤ i ≤ t , where a switch is the replacement of a pair of edges u v and w z on a Hamiltonian cycle with the edges u w and v z of G, given that u w and v z did not appear on the cycle. We show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete, settling an open question posed by Ito et al. (2011) and van den Heuvel (2013). More precisely, we show that the Hamiltonian cycle reconfiguration problem is PSPACE-complete for chordal bipartite graphs, strongly chordal split graphs, and bipartite graphs with maximum degree 6. Bipartite permutation graphs form a proper subclass of chordal bipartite graphs, and unit interval graphs form a proper subclass of strongly chordal graphs. On the positive side, we show that, for any two Hamiltonian cycles of a bipartite permutation graph and a unit interval graph, there is a sequence of switches transforming one cycle to the other, and such a sequence can be obtained in linear time.

BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

2009 ◽  
Vol 10 (03) ◽  
pp. 205-217 ◽  
Author(s):  
WEIZHEN GU ◽  
KIRSTI WASH
Keyword(s):  
Maximum Degree ◽  
Domination Number ◽  
Permutation Graph ◽  
Original Graph ◽  
Permutation Graphs ◽  
Positive Integers

For a graph G with n vertices and a permutation α on V(G), a permutation graph Pα(G) is obtained from two identical copies of G by adding an edge between v and α(V) for any v ϵ V(G). Let γ(G) be the domination number of a graph G. It has been shown that γ(G) ≤ γ(Pα(G) ≤ 2γ(G) for any permutation α on V(G). In this paper, we investigate specific graphs for which there exists a permutation α such that γ(Pα(G)) ≻ γ(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation α for any two positive integers a and b with a ≤ b ≤ 2a, to have γ(G) = a and γ(Pα(G)) = b.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

Finding a maximum matching in a permutation graph

1995 ◽  
Vol 32 (8) ◽  
pp. 779-792 ◽  
Author(s):  
Chongkye Rhee ◽  
Y. Daniel Liang
Keyword(s):  
Maximum Matching ◽  
Permutation Graph

scholarly journals Games on interval and permutation graph representations

2016 ◽  
Vol 609 ◽  
pp. 87-103
Author(s):  
Jessica Enright ◽  
Lorna Stewart
Keyword(s):  
Permutation Graph ◽  
Graph Representations

Fully dynamic algorithms for permutation graph coloring

1997 ◽  
Vol 63 (1-2) ◽  
pp. 37-55
Author(s):  
Zoran Ivković ◽  
Ramnath Sarnath ◽  
Sivaprakasam Sunder
Keyword(s):  
Graph Coloring ◽  
Permutation Graph ◽  
Dynamic Algorithms

MAXIMUM INDEPENDENT SET OF A PERMUTATION GRAPH IN K TRACKS

1993 ◽  
Vol 03 (03) ◽  
pp. 291-304 ◽  
Author(s):  
D.T. LEE ◽  
MAJID SARRAFZADEH
Keyword(s):  
Independent Set ◽  
Maximum Size ◽  
Interval Graph ◽  
Vertical Line ◽  
Maximum Independent Set ◽  
Permutation Graph ◽  
End Points

A maximum weighted independent set of a permutation graph is a maximum subset of noncrossing chords in a matching diagram (i.e., a set Φ of chords with end-points on two horizontal lines). The problem of finding, among all noncrossing subsets of Φ with density at most k, one with maximum size is considered, where the density of a subset is the maximum number of chords crossing a vertical line and k is a given parameter. A Θ(n log n) time and Θ(n) space algorithm, for solving the problem with n chords, is proposed. As an application, we solve the problem of finding, among all proper subsets with density at most k of an interval graph, one with maximum number of intervals.

Finding a chain graph in a bipartite permutation graph

2016 ◽  
Vol 116 (9) ◽  
pp. 569-573 ◽  
Author(s):  
Masashi Kiyomi ◽  
Yota Otachi
Keyword(s):  
Chain Graph ◽  
Permutation Graph ◽  
A Chain
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