Coloring permutation graphs in parallel

Abstract

A coloring of a graph G is an assignment of colors to its vertices so that no two adjacent vertices have the same color. We study the problem of coloring permutation graphs using certain properties of the lattice representation of a permutation and relationships between permutations, directed acyclic graphs and rooted trees having speciÿc key properties. We propose an e cient parallel algorithm which colors an n-node permutation graph in O(log 2 n) time using O(n 2 =log n) processors on the CREW PRAM model. Speciÿcally, given a permutation we construct a tree T * [ ], which we call coloring-permutation tree, using certain combinatorial properties of . We show that the problem of coloring a permutation graph is equivalent to ÿnding vertex levels in the coloring-permutation tree. ?

The Number of Spanning Trees in Kn-Complements of Quasi-Threshold Graphs

“…It has been shown that the classes of perfect graphs, namely complement reducible graphs, or socalled cographs, and permutation graphs, have nice structural and algorithmic properties: a cograph admits a unique tree representation, up to isomorphism, called a cotree [11] (note that the class of cographs contain the classes of quasi-threshold and threshold graphs), while a permutation graph G[π] can be transformed into a directed acyclic graph and, then, into a rooted tree by exploiting the inversion relation on the elements of the permutation π [14].…”

Encoding watermark numbers as reducible permutation graphs using self-inverting permutations

“…We say that an element i of the permutation π dominates the element j if i > j and π −1 i < π −1 j . An element i directly dominates (or, for short, didominates) the element j if i dominates j and there exists no element k in π such that i dominates k and k dominates j [25]. For example, in the permutation π = (8, 3, 2, 7, 1, 9, 6, 5, 4), the element 7 dominates the elements 1, 6, 5, 4 and it directly dominates the elements 1, 6.…”

Algorithmic techniques for encoding permutations and permutation graphs for watermarking software, image, audio, and text

“…We say that an element i of a permutation over the set N n dominates the element j if i > j and −1 i < −1 j . An element i directly dominates (or d-dominates, for short) the element j if i dominates j and there exists no element k in such that i dominates k and k dominates j [94]. For example, in the permutation = (8; 3; 2; 7; 1; 9; 6; 5; 4), the element 7 dominates the elements 1; 6; 5; 4 and directly dominates the elements 1; 6.…”