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Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 111 (12 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
A simple lineartime modular decomposition algorithm for graphs, using order extension
, 2004
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The NTree: A Two Dimension Partial Order for Protection Groups
 ACM TRANSACTIONS ON COMPUTER SYSTEMS
, 1988
"... The benefits of providing access control with groups of users rather than individuals as the unit of granularity are well known. These benefits are enhanced if the groups are organized in a subgroup partial order. A class of such partial orders called ntrees is de#ned by using a forest of rooted tr ..."
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Cited by 17 (12 self)
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The benefits of providing access control with groups of users rather than individuals as the unit of granularity are well known. These benefits are enhanced if the groups are organized in a subgroup partial order. A class of such partial orders called ntrees is de#ned by using a forest of rooted trees or inverted rooted trees as basic partial orders and combining these by refinement. Refinement explodes an existing group into a partially ordered ntree of new groups while maintaining the same relationship between each new group and the nonexploded groups that the exploded group had. Examples are discussed to show the practical significance of ntrees and the refinement operation. It is shown that ntrees can be represented by assigning a pair of integers called lrvalues to each group so that g is a subgroup of h if only if l#g# # l#h# and r#g# # r#h#. Refinement allows a complex ntree to be developed incrementally in a topdown manner and is useful for the initial definition of an ntree as we...
NC algorithms for comparability graphs, interval graphs, and unique perfect matching
 Proc. 5th Conf. Found. Software Technology and Theor. Comput. Sci., volume 206 of Lect. Notes in Comput. Sci
, 1985
"... Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interva ..."
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Cited by 12 (0 self)
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Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for nding a maximum matching in an incomparability graph. 1
Coloring Permutation Graphs in Parallel
, 2002
"... A coloring of a graph G is an assignment ofcol]B to its vertices so that no two adjacent vertices have the samecol.]B study theproblP ofcolxxPpermutation graphs using certain properties of thele.Pq representation of a permutation andrel#B[.bEPq between permutations, directed acycld graphs and roo ..."
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Cited by 5 (5 self)
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A coloring of a graph G is an assignment ofcol]B to its vertices so that no two adjacent vertices have the samecol.]B study theproblP ofcolxxPpermutation graphs using certain properties of thele.Pq representation of a permutation andrel#B[.bEPq between permutations, directed acycld graphs and rooted trees having speci#c key properties.We propose an e#cient paralnt allnt.P which colh. an nnode permutation graph inO(lO =l.
Recognizing Immediacy in an NTree Hierarchy and its Application to Protection Groups
 IEEE Transactions on Software Engineering
, 1989
"... The benefits of providing access control with groups of users as the unit of granularity are well known. These benefits are enhanced if the groups are organized in a hierarchy (partial order) by the subgroup relation #, where g # h signifies that every member of group g is thereby also a member of g ..."
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Cited by 2 (1 self)
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The benefits of providing access control with groups of users as the unit of granularity are well known. These benefits are enhanced if the groups are organized in a hierarchy (partial order) by the subgroup relation #, where g # h signifies that every member of group g is thereby also a member of group h. It is often useful to distinguish the case when g is an immediate subgroup of h, that is when g # h and there is no group k such that g # k # h. The class of partial orders called ntrees was recently defined by using rooted trees and inverted rooted trees as basic partial orders and combining these recursively by refinement [12]. It has been shown that ntrees arise naturally in many practical situations and they have a simple representation. Any ntree hierarchy can be expressed as the intersection of two linear orderings. So it is possible to assign a pair of integers l[x] and r[x] to each group x such that g # h if and only if l[g] &le; l[h] and r[g] &le; r[h]. In this paper we show ho...
Extremal Problems on Edge_Colorings, . . . CYCLE SPECTRA OF GRAPHS
, 2010
"... We study problems in extremal graph theory with respect to edgecolorings, independent sets, and cycle spectra. In Chapters 2 and 3, we present results in Ramsey theory, where we seek Ramsey host graphs with small maximum degree. In Chapter 4, we study a Ramseytype problem on edgelabeled trees, whe ..."
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We study problems in extremal graph theory with respect to edgecolorings, independent sets, and cycle spectra. In Chapters 2 and 3, we present results in Ramsey theory, where we seek Ramsey host graphs with small maximum degree. In Chapter 4, we study a Ramseytype problem on edgelabeled trees, where we seek subtrees that have a small number of pathlabels. In Chapter 5, we examine parity edgecolorings, which have connections to additive combinatorics and the minimum dimension of a hypercube in which a tree embeds. In Chapter 6, we prove results on the chromatic number of circle graphs with clique number at most 3. The tournament analogue of an independent set is an acyclic set. In Chapter 7, we present results on the size of maximum acyclic sets in kmajority tournaments. In Chapter 8, we prove a lower bound on the size of the cycle spectra of Hamiltonian graphs.