Results 11  20
of
93
Graph Decompositions and Factorizing Permutations
 Discrete Mathematics and Theoretical Computer Science
, 1997
"... A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propo ..."
Abstract

Cited by 13 (9 self)
 Add to MetaCart
(Show Context)
A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu [9, 8] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [7] for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.
Structure and Stability Number of Chair, CoP and GemFree Graphs Revisited
, 2002
"... The P 4 is the induced path with vertices a; b; c; d and edges ab; bc; cd. The chair (coP, gem) has a fifth vertex adjacent to b (a and b, a; b; c and d respectively). We give a complete structure description of prime chair, coP and gemfree graphs which implies bounded clique width for this gra ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
The P 4 is the induced path with vertices a; b; c; d and edges ab; bc; cd. The chair (coP, gem) has a fifth vertex adjacent to b (a and b, a; b; c and d respectively). We give a complete structure description of prime chair, coP and gemfree graphs which implies bounded clique width for this graph class. It is known that this has some nice consequences; very roughly speaking, every problem expressible in a certain kind of Monadic Second Order Logic (quantifying only over vertex set predicates) can be solved efficiently for graphs of bounded clique width. In particular, we obtain linear time for the problems Vertex Cover, Maximum Weight Stable Set (MWS), Maximum Weight Clique, Steiner Tree, Domination and Maximum Induced Matching on chair, coP and gemfree graphs and a slightly larger class of graphs. This drastically improves a recently published O(n ) time bound for Maximum Stable Set on buttery, chair, coP and gemfree graphs.
C.: Drawing graphs using modular decomposition
 In Graph Drawing. Volume LNCS 3843
, 2005
"... Abstract. In this paper we present an algorithm for drawing an undirected graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by traversing the modular decomposition tree of the input graph G on n vertices and m edges, in a bottom ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we present an algorithm for drawing an undirected graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by traversing the modular decomposition tree of the input graph G on n vertices and m edges, in a bottomup fashion until it reaches the root of the tree, while at the same time intermediate drawings are computed. In order to achieve aesthetically pleasing results, we use grid and circular placement techniques, and utilize an appropriate modification of a wellknown spring embedder algorithm. It turns out, that for some classes of graphs, our algorithm runs in O(n+m) time, while in general, the running time is bounded in terms of the processing time of the spring embedder algorithm. The result is a drawing that reveals the structure of the graph G and preserves certain aesthetic criteria. 1
Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs
 ALGORITHMICA
, 2009
"... This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposit ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs.
Algebraic Operations on PQ Trees and Modular Decomposition Trees
, 2005
"... Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise o ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise on a variety of other combinatorial structures. We describe natural operators on partitive set families, give algebraic identities for manipulating them, and describe efficient algorithms for evaluating them. We use these results to obtain new time bounds for finding the common intervals of a set of permutations, finding the modular decomposition of an edgecolored graphs (also known as a twostructure), finding the PQ tree of a matrix when a consecutiveones arrangement is given, and finding the modular decomposition of a permutation graph when its permutation realizer is given.
Ordered Vertex Partitioning
, 2000
"... A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modulardecomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In t ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modulardecomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n+mlogn)algorithm that uses this duality to find both a transitive orientation and the modular decomposition. Though the running time is not optimal, this algorithm is much simpler than any previous algorithms that are not Ω(n²). The bestknown time bounds for the problems are O(n + m), but they involve sophisticated techniques.
Gem and CoGemFree Graphs Have Bounded CliqueWidth
, 2003
"... The P 4 is the induced path of four vertices. The gem consists of a P 4 with an additional universal vertex being completely adjacent to the P 4 , and the cogem is its complement graph. Gem and cogemfree graphs generalize the popular class of cographs (i. e. P 4 free graphs). The tree structure ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
The P 4 is the induced path of four vertices. The gem consists of a P 4 with an additional universal vertex being completely adjacent to the P 4 , and the cogem is its complement graph. Gem and cogemfree graphs generalize the popular class of cographs (i. e. P 4 free graphs). The tree structure and algebraic generation of cographs has been crucial for several concepts of graph decomposition such as modular and homogeneous decomposition. Moreover, it is fundamental for the recently introduced concept of cliquewidth of graphs which extends the famous concept of treewidth. It is wellknown that the cographs are exactly those graphs of cliquewidth at most 2.
Optimal linear arrangement of interval graphs
 in Proceedings of the 31st International Symposium on Mathematical Foundations of Computer Science (MFCS 2006
"... Abstract. We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NPhard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NPhard. The s ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We study the optimal linear arrangement (OLA) problem on interval graphs. Several linear layout problems that are NPhard on general graphs are solvable in polynomial time on interval graphs. We prove that, quite surprisingly, optimal linear arrangement of interval graphs is NPhard. The same result holds for permutation graphs. We present a lower bound and a simple and fast 2approximation algorithm based on any interval model of the input graph. 1
SPLITPERFECT GRAPHS: CHARACTERIZATIONS AND ALGORITHMIC USE
, 2004
"... Two graphs G and H with the same vertex set V are P4isomorphic if every four vertices {a, b, c, d} ⊆V induce a chordless path (denoted by P4) inG if and only if they induce a P4 in H. We call a graph splitperfect if it is P4isomorphicto a split graph (i.e., a graph being partitionable into a cl ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Two graphs G and H with the same vertex set V are P4isomorphic if every four vertices {a, b, c, d} ⊆V induce a chordless path (denoted by P4) inG if and only if they induce a P4 in H. We call a graph splitperfect if it is P4isomorphicto a split graph (i.e., a graph being partitionable into a clique and a stable set). This paper characterizes the new class of splitperfect graphs using the concepts of homogeneous sets and pconnected graphs and leads to a linear time recognition algorithm for splitperfect graphs, as well as efficient algorithms for classical optimization problems on splitperfect graphs based on the primeval decomposition of graphs. The optimization results considerably extend previous ones on smaller classes such as P4sparse graphs, P4lite graphs, P4laden graphs, and (7,3)graphs. Moreover, splitperfect graphs form a new subclass of brittle graphs containing the superbrittle graphs for which a new characterization is obtained leading to linear time recognition.