Generating Chordal Graphs Included in Given Graphs (original) (raw)

Enumeration of the perfect sequences of a chordal graph

Theoretical Computer Science, 2010

A graph is chordal if and only if it has no chordless cycle of length more than three. The set of maximal cliques in a chordal graph admits special tree structures called clique trees. A perfect sequence is a sequence of maximal cliques obtained by using the reverse order of repeatedly removing the leaves of a clique tree. This paper addresses the problem of enumerating all the perfect sequences. Although this problem has statistical applications, no efficient algorithm has been proposed. There are two difficulties with developing this type of algorithm. First, a chordal graph does not generally have a unique clique tree. Second, a perfect sequence can normally be generated by two or more distinct clique trees. Thus it is hard using a straightforward algorithm to generate perfect sequences from each possible clique tree. In this paper, we propose a method to enumerate perfect sequences without constructing clique trees. As a result, we have developed the first polynomial delay algorithm for dealing with this problem. In particular, the time complexity of the algorithm on average is O(1) for each perfect sequence.

Efficient Enumeration of All Chordless Cycles in Graphs

In a finite undirected simple graph, a {\it chordless cycle} is an induced subgraph which is a cycle. We propose two algorithms to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithms have the advantage of finding each chordless cycle only once. To ensure this, we introduced the concepts of vertex labeling and initial valid vertex triplet. To guarantee that the expansion of a given chordless path will always lead to a chordless cycle, we use a breadth-first search in a subgraph obtained by the elimination of many of the vertices from the original graph. The resulting algorithm has time complexity mathcalO(n+m)\mathcal{O}(n + m)mathcalO(n+m) in the output size, where nnn is the number of vertices and mmm is the number of edges.

Clique tree generalization and new subclasses of chordal graphs

Discrete Applied Mathematics, 2002

The notion of a clique tree plays a central role in obtaining an intersection graph representation of a chordal graph. In this paper, we introduce a new structure called the reduced clique hypergraph of a chordal graph. Unlike a clique tree, the reduced clique hypergraph is a unique structure associated with a chordal graph. We show that all clique trees of a chordal graph can be obtained from the reduced clique hypergraph; thus the reduced clique hypergraph can be thought of as a generalization of the notion of a clique tree. We then link the reduced clique hypergraph notion to minimal vertex separators of chordal graph by proving a structure theorem which shows that the edges of the reduced clique hypergraph are in one-one correspondence with the minimal vertex separators. We also show that a closed-form formula for the number of clique trees of a chordal graph can be derived using these results. Using an algorithmic characterization of minimal vertex separators, we obtain e cient algorithms to compute the reduced clique hypergraph and to count the number of clique trees of a chordal graph. Finally, guided by the reduced clique hypergraph structure we propose a few new subclasses of chordal graphs and relate them to each other and the existing subclasses.

Chordal graphs and their clique graphs

Graph-Theoretic Concepts in Computer Science, 1995

In this paper, we present a new structure for chordal graph. We have also given the algorithm for MCS(Maximal Cardinality Search) and lexicographic BFS(Breadth First Search) which is used in two linear time and space algorithm. Also we discuss how to build a clique tree of a chordal graph and the other is simple recognition procedure of chordal graphs.

Efficient Enumeration of Chordless Cycles

arXiv: Data Structures and Algorithms, 2013

In a finite undirected simple graph, a {\it chordless cycle} is an induced subgraph which is a cycle. We propose two algorithms to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithms have the advantage of finding each chordless cycle only once. To ensure this, we introduced the concepts of vertex labeling and initial valid vertex triplet. To guarantee that the expansion of a given chordless path will always lead to a chordless cycle, we use a breadth-first search in a subgraph obtained by the elimination of many of the vertices from the original graph. The resulting algorithm has time complexity mathcalO(n+m)\mathcal{O}(n + m)mathcalO(n+m) in the output size, where nnn is the number of vertices and mmm is the number of edges.

D M ] 1 1 N ov 2 01 8 Generating subgraphs in chordal graphs

2018

A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight. For every graphG, the set of weight functions w such thatG is w-well-covered is a vector space, denoted WCW (G). Let B be a complete bipartite induced subgraph of G on vertex sets of bipartition BX and BY . Then B is generating if there exists an independent set S such that S ∪ BX and S ∪ BY are both maximal independent sets of G. In the restricted case that a generating subgraph B is isomorphic to K1,1, the unique edge in B is called a relating edge. Generating subgraphs play an important role in finding WCW (G). Deciding whether an input graph G is well-covered is co-NP-complete. Hence, finding WCW (G) is co-NP-hard. Deciding whether an edge is relating is NP-complete. Therefore, deciding whether a subgraph is generating is NP-complete as well. A graph ...

Determining what sets of trees can be the clique trees of a chordal graph

Journal of the Brazilian Computer Society, 2011

Chordal graphs have characteristic tree representations, the clique trees. The problems of finding one or enumerating them have already been solved in a satisfactory way. In this paper, the following related problem is studied: given a family "Equation missing" of trees, all having the same vertex set V, determine whether there exists a chordal graph whose set of clique trees equals "Equation missing". For that purpose, we undertake a study of the structural properties, some already known and some new, of the clique trees of a chordal graph and the characteristics of the sets that induce subtrees of every clique tree. Some necessary and sufficient conditions and examples of how they can be applied are found, eventually establishing that a positive or negative answer to the problem can be obtained in polynomial time. If affirmative, a graph whose set of clique trees equals "Equation missing" is also obtained. Finally, all the chordal graphs with set of c...

Dynamic algorithms for chordal and interval graphs

2001

We present the first dynamic algorithm that maintains a clique tree representation of a chordal graph and supports the following operations: (1) query whether deleting or inserting an arbitrary edge preserves chordality, (2) delete or insert an arbitrary edge, provided it preserves chordality. We give two implementations. In the first, each operation runs in O( n) time, where n is the number of vertices. In the second, an insertion query runs in O(log2 n) time, an insertion in O(n) time, a deletion query in O(n) time, and a deletion in O(n log n) time. We also introduce the clique-separator graph representation of a chordal graph, which provides significantly more information about the graph's structure than the well-known clique tree representation. We present fundamental properties of the clique-separator graph and additional properties when the input graph is interval. We then introduce the train tree representation of interval graphs and use it to decide whether there is a c...

On listing, sampling, and counting the chordal graphs with edge constraints

Theoretical Computer Science, 2010

We discuss the problems to list, sample, and count the chordal graphs with edge constraints. The edge constraints are given as a pair of graphs one of which contains the other and one of which is chordal, and the objects we look at are the chordal graphs contained in one and containing the other. This setting is a natural generalization of chordal completions and deletions. For the listing problem, we give an efficient algorithm running in amortized polynomial time per output with polynomial space. For the sampling problem, we give an instance for which a natural Markov chain suffers from an exponential mixing time. For the counting problem, we show some #P-completeness results. These results provide a unified viewpoint from algorithms theory to problems arising from various areas such as statistics, data mining, and numerical computation.