Perfect Elimination Ordering Research Articles

LexBFS-orderings of distance-hereditary graphs with application to the diametral pair problem

For an undirected graph G the kth power G k of G is the graph with the same vertex set as G where two vertices are adjacent iff their distance is at most k in G. In this paper we prove that every LexBFS-ordering of a distance-hereditary graph is both a common perfect elimination ordering of all even powers and a common semi-simplicial ordering of all powers of this graph. Moreover, we characterize those distance-hereditary graphs by forbidden subgraphs for which every LexBFS-ordering of the graph is a common perfect elimination ordering of all powers. As an application we present an algorithm which computes the diameter and a diametral pair of vertices of a distance-hereditary graph in linear time.

Lexbfs-orderings and powers of hhd-free graphs *

The k-th power of a graph G is the graph with the same vertex set as G where two vertices are adjacent iff their distance in G is at most k. We investigate LexBFS-orderings of HHD-free graphsi.e., the graphs where each cycle of length at least five has two chords, with respect to their powers, It is well-known that any LexBFS-ordering of a HHD-free graph is a so-called semisimplicial ordering. We show, that any LexBFS-ordering of a HHD-free graph is a common semisimplicial ordering of all its odd powers. Furthermore, we characterize those HHD-free graphs by forbidden subgraphs for which any LexBFS-ordering of the initial graph is a common perfect elimination ordering of all odd or even powers

Minimal vertex separators of chordal graphs

Chordal graphs form an important and widely studied subclass of perfect graphs. The set of minimal vertex separators constitute an unique class of separators of a chordal graph and capture the structure of the graph. In this paper, we explore the connection between perfect elimination orderings of a chordal graph and its minimal vertex separators. Specifically, we prove a characterization of these separators in terms of the monotone adjacency sets of the vertices of the graph, numbered by the maximum cardinality search (MCS) scheme. This leads to a simple linear-time algorithm to identify the minimal vertex separators of a chordal graph using the MCS scheme. We also introduce the notion of multiplicity of a minimal vertex separator which indicates the number of different pairs of vertices separated by it. We prove a useful property of the lexicographic breadth first scheme (LBFS) that enables us to determine the multiplicities of minimal vertex separators of a chordal graph.

Computation of a (canonical) doubly perfect elimination ordering of a doubly chordal graph

The class ofdoubly chordal graphs is a subclass ofchordal graphs and a superclass ofstrongly chordal graphs, which arise in so many application areas. Many optimization problems like domination and Steiner tree are NP-complete on chordal graphs but can be solved in polynomial time on doubly chordal graphs. The central to designing efficient algorithms for doulby chordal graphs is the concept of(canonical) doubly perfect elimination orderings. We present linear time algorithms to compute a(canonical) doubly perfect elimination ordering of adoubly chordal graph.

Matching and multidimensional matching in chordal and strongly chordal graphs

Chordal graphs are graphs with the property that each cycle of length greater than 3 has two non-consecutive vertices that are joined by an edge. An important subclass of chordal graphs are strongly chordal graphs (Farber, 1983). Chordal graphs appear for example in the design of acyclic data base schemes (Beeri et al., 1983). In this paper we study the computational complexity (both sequential and parallel) of the maximum matching problem for chordal and strongly chordal graphs. We show that there is a linear-time greedy algorithm for a maximum matching in a strongly chordal graph provided a strongly perfect elimination ordering is known. This algorithm can also be turned into a parallel algorithm. The technique used can also be extended to the perfect multidimensional matching for chordal and strongly chordal graphs yielding the first polynomial time algorithms for these classes of graphs (the multidimensional matching is NP-complete in general).

The algorithmic use of hypertree structure and maximum neighbourhood orderings

The use of (generalized) tree structure in graphs is one of the main topics in the field of efficient graph algorithms. The well-known partial k-tree (resp. treewidth) approach belongs to this kind of research and bases on a tree structure of constant-size bounded maximal cliques. Without size bound on the cliques this tree structure of maximal cliques characterizes chordal graphs which are known to be important also in connection with relational database schemes where hypergraphs with tree structure (acyclic hypergraphs) and their elimination orderings (perfect elimination orderings for chordal graphs, Graham-reduction for acyclic hypergraphs) are studied. We consider here graphs with a tree structure which is dual (in the sense of hypergraphs) to that one of chordal graphs (therefore we call these graphs dually chordal). The corresponding vertex elimination orderings of these graphs are the maximum neighbourhood orderings. These orderings were studied recently in several papers and some of the algorithmic consequences of such orderings are given. The aim of this paper is a systematic treatment of the algorithmic use of maximum neighhourhood orderings. These orderings are useful especially for dominating-like problems (including Steiner tree) and distance problems. Many problems efficiently solvable for strongly chordal and doubly chordal graphs remain efficiently solvable for dually chordal graphs too. Our results on dually chordal graphs not only generalize, but also improve and extend the corresponding results on strongly chordal and doubly chordal graphs, since a maximum neighbourhood ordering (if it exists) can be constructed in linear time and we consequently use the underlying structure properties of dually chordal graphs closely connected to hypergraphs. Furthermore, a collection of problems remaining NP -complete on dually chordal graphs is given.

Some characterizations of doubly chordal graphs

Many optimization problems like domination and Steiner tree are NP-complete onchordal graphs but can be solved in polynomial time ondoubly chordal graphs. Investigating properties of doubly chordal graphs probably help to design efficient algorithms for the graphs. We present some characterizations of doubly chordal graphs, which are based on clique matrices and neighborhood matrices. It is also mentioned how adoubly perfect elimination ordering of a doubly chordal graph can be computed from the results.

LexBFS-orderings and powers of chordal graphs

For an undirected graph G the kth power G k of G is the graph with the same vertex set as G where two vertices are adjacent iff their distance is at most k in G. In this paper we prove that any LexBFS-ordering of a chordal graph is a common perfect elimination ordering of all odd powers of this graph. Moreover, we characterize those chordal graphs by forbidden isometric subgraphs for which any LexBFS-ordering of the graph is a common perfect elimination ordering of all powers. For MCS-orderings of chordal graphs the situation is worse: even for trees MCS does not give a common perfect elimination ordering of powers.

Perfect elimination orderings of chordal powers of graphs

Let G = ( V, E) be a finite undirected connected graph. We show that there is a common perfect elimination ordering of all powers of G which represent chordal graphs. Consequently, if G and all of its powers are chordal then all these graphs admit a common perfect elimination ordering. Such an ordering can be computed in O(| V| · | E|) time using a generalization of the Tarjan and Yannakakis' Maximum Cardinality Search.

Efficient Parallel Algorithms for Chordal Graphs

We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadth-first search tree and a depth-first search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to our results is an efficient parallel algorithm for finding a perfect elimination ordering.

New linear time algorithms for generating perfect elimination orderings of chordal graphs

We propose two linear time algorithms: maximum cardinality depth-first search (MCDFS) and maximum cardinality breadth-first search (MCBFS), which are variants of depth-first search and breadth-first search, respectively, to generate perfect elimination orderings of chordal graphs.

A clique tree algorithm for partitioning a chordal graph into transitive subgraphs

A partitioning problem on chordal graphs that arises in the solution of sparse triangular systems of equations on parallel computers is considered. Roughly the problem is to partition a chordal graph G into the fewest transitively orientable subgraphs over all perfect elimination orderings of G, subject to a certain precedence relationship on its vertices. In earlier work, a greedy scheme that solved the problem by eliminating a largest subset of vertices at each step was described, and an algorithm implementing the scheme in time and space linear in the number of edges of the graph was provided. A more efficient greedy scheme, obtained by representing the chordal graph in terms of its maximal cliques, is described here. The new greedy scheme eliminates, in a specified order, a largest set of “persistent leaves,” a subset of the leaf cliques in the current graph, at each step. Several new results about minimal vertex separators in chordal graphs, and in particular, the concept of a critical separator of a leaf clique, are employed to prove that the new scheme solves the partitioning problem. We provide an algorithm implementing the scheme in time and space linear in the size of the clique tree. We anticipate that a critical separator of a leaf clique may be a useful concept in other problems on chordal graphs.

A parallel algorithm for computing Steiner trees in strongly chordal graphs

We present an efficient parallel algorithm for the computation of a minimum Steiner tree for any strongly chordal graph. The algorithm works in O(log 2 n) time and uses a linear number of processors provided a strongly perfect elimination ordering is given.

Partitioning a chordal graph into transitive subgraphs for parallel sparse triangular solution

A recent approach for solving sparse triangular systems of equations on massively parallel computers employs a factorization of the triangular coefficient matrix to obtain a representation of its inverse in product form. The number of general communication steps required by this approach is proportional to the number of factors in the factorization. The triangular matrix can be symmetrically permuted to minimize the number of factors over suitable classes of permutations, and thereby the complexity of the parallel algorithm can be minimized. Algorithms for minimizing the number of factors over several classes of permutations have been considered in earlier work. Let F = L + L T denote the symmetric filled matrix corresponding to a Cholesky factor L, and let G F denote the adjacency graph of F. We consider the problem of minimizing the number of factors over all permutations which preserve the structure of G F . The graph model of this problem is to partition the vertices G F into the fewest transitively closed subgraphs over all perfect elimination orderings while satisfying a certain precedence relationship. The solution to this chordal-graph partitioning problem can be described by a greedy scheme which eliminates a largest permissible subgraph at each step. Further, the subgraph eliminated at each step can be characterized in terms of lengths of chordless paths in the current elimination graph. This solution relies on several results concerning transitive perfect elimination orderings introduced in this paper. We describe a partitioning algorithm with O (∣ V∣ + ∣ E∣) time and space complexity.

Some aspects of the semi-perfect elimination

Several efficient algorithms have been proposed to construct a perfect elimination ordering of the vertices of a chordal graph. We study the behaviour of two of these algorithms in relation to a new concept, namely the semi-perfect elimination ordering, which provides a natural generalization of chordal graphs.

Some aspects of perfect elimination orderings in chordal graphs

This paper studies properties of perfect elimination orderings in chordal graphs. Specific connections to convex subsets and quasiconcave functions in a graph are discussed. Several new schemes for generating all perfect elimination orderings are investigated and related to existing schemes.

Algorithmic Aspects of Vertex Elimination on Graphs

We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse symmetric positive definite systems of linear equations. We give a new linear-time algorithm to calculate the fill-in produced by any elimination ordering, and we give two new related algorithms for finding orderings with special properties. One algorithm, based on breadth-first search, finds a perfect elimination ordering, if any exists, in $O(n + e)$ time, if the problem graph has n vertices and e edges. An extension of this algorithm finds a minimal (but not necessarily minimum) ordering in $O(ne)$ time. We conjecture that the problem of finding a minimum ordering is NP-complete