Chordal Subgraphs Research Articles

Maximum chordal subgraphs of random graphs

AbstractWe find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$ , for $p=\mathrm{const}$ and $p=n^{-\alpha +o(1)}$ .

Maximal chordal subgraphs

AbstractA chordal graph is a graph with no induced cycles of length at least $4$ . Let $f(n,m)$ be the maximal integer such that every graph with $n$ vertices and $m$ edges has a chordal subgraph with at least $f(n,m)$ edges. In 1985 Erdős and Laskar posed the problem of estimating $f(n,m)$ . In the late 1980s, Erdős, Gyárfás, Ordman and Zalcstein determined the value of $f(n,n^2/4+1)$ and made a conjecture on the value of $f(n,n^2/3+1)$ . In this paper we prove this conjecture and answer the question of Erdős and Laskar, determining $f(n,m)$ asymptotically for all $m$ and exactly for $m \leq n^2/3+1$ .

Proximity Search for Maximal Subgraph Enumeration

This paper proposes a new general technique for maximal subgraph enumeration which we call proximity search, whose aim is to design efficient enumeration algorithms for problems that could not be solved by existing frameworks. To support this claim and illustrate the technique we include output-polynomial algorithms for several problems for which output-polynomial algorithms were not known, including the enumeration of maximal bipartite subgraphs, maximal -degenerate subgraphs (for bounded ), maximal induced chordal subgraphs, and maximal induced trees. Using known techniques, such as reverse search, the space of all maximal solutions induces an implicit directed graph called “solution graph” or “supergraph,” and solutions are enumerated by traversing it; however, nodes in this graph can have exponential out-degree, thus requiring exponential time to be spent on each solution. The novelty of proximity search is a formalization that allows us to define a better solution graph, and a technique, which we call canonical reconstruction, by which we can exploit the properties of given problems to build such graphs. This results in solution graphs whose nodes have significantly smaller (i.e., polynomial) out-degree with respect to existing approaches, but that remain strongly connected, so that all solutions can be enumerated in polynomial delay by a traversal. A drawback of this approach is the space required to keep track of visited solutions, which can be exponential; we further propose a technique to induce a parent-child relationship among solutions and achieve polynomial space when suitable conditions are met.

A new augmentation based algorithm for extracting maximal chordal subgraphs

A graph is chordal if every cycle of length greater than three contains an edge between non-adjacent vertices. Chordal graphs are of interest both theoretically, since they admit polynomial time solutions to a range of NP-hard graph problems, and practically, since they arise in many applications including sparse linear algebra, computer vision, and computational biology. A maximal chordal subgraph is a chordal subgraph that is not a proper subgraph of any other chordal subgraph. Existing algorithms for computing maximal chordal subgraphs depend on dynamically ordering the vertices, which is an inherently sequential process and therefore limits the algorithms’ parallelizability.In this paper we explore techniques to develop a scalable parallel algorithm for extracting a maximal chordal subgraph. We demonstrate that an earlier attempt at developing a parallel algorithm may induce a non-optimal vertex ordering and is therefore not guaranteed to terminate with a maximal chordal subgraph. We then give a new algorithm that first computes and then repeatedly augments a spanning chordal subgraph. After proving that the algorithm terminates with a maximal chordal subgraph, we then demonstrate that this algorithm is more amenable to parallelization and that the parallel version also terminates with a maximal chordal subgraph. That said, the complexity of the new algorithm is higher than that of the previous parallel algorithm, although the earlier algorithm computes a chordal subgraph which is not guaranteed to be maximal.We experimented with our augmentation-based algorithm on both synthetic and real-world graphs. We provide scalability results and also explore the effect of different choices for the initial spanning chordal subgraph on both the running time and on the number of edges in the maximal chordal subgraph.

A Parallel Graph Sampling Algorithm for Analyzing Gene Correlation Networks

Abstract Effcient analysis of complex networks is often a challenging task due to its large size and the noise inherent in the system. One popular method of overcoming this problem is through graph sampling, that is extracting a representative subgraph from the larger network. The accuracy of the sample is validated by comparing the combinatorial properties of the subgraph and the original network. However, there has been little study in comparing networks based on the applications that they represent. Furthermore, sampling methods are generally applied agnostically, without mapping to the requirements of the underlying analysis. In this paper,we introduce a parallel graph sampling algorithm focusing on gene correlation networks. Densely connected subgraphs indicate important functional units of gene products. In our sampling algorithm, we emphasize maintaining highly connected regions of the network through parallel sampling based on extracting the maximal chordal subgraph of the network. We validate our methods by comparing both combinatorial properties and functional units of the subgraphs and larger networks. Our results show that even with significant reduction of the network (on average 20% to 40%), we obtain reliable samplings and many of the relevant combinatorial and functional properties are retained in the subgraphs.

On the complexity of some subgraph problems

We study the complexity of the problem of deciding the existence of a spanning subgraph of a given graph, and of that of finding a maximum (weight) such subgraph. We establish some general relations between these problems, and we use these relations to obtain new N P -completeness results for maximum (weight) spanning subgraph problems from analogous results for existence problems and from results in extremal graph theory. On the positive side, we provide a decomposition method for the maximum (weight) spanning chordal subgraph problem that can be used, e.g., to obtain a linear (or O ( n log n ) ) time algorithm for such problems in graphs with vertex degree bounded by 3.

A vertex incremental approach for maintaining chordality

For a chordal graph G = ( V , E ) , we study the problem of whether a new vertex u ∉ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv , a characterization of the set of edges incident to u that also must be added to G along with uv . We propose a data structure that can compute and add each such set in O ( n ) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O ( nm ) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is on-line in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently.

Maximal sub-triangulation in pre-processing phylogenetic data

In order to help infer an evolutionary tree (phylogeny) from experimental data, we propose a new method for pre-processing the corresponding dissimilarity matrix, which is related to the property that the distance matrix of a phylogeny (called an additive matrix) describes a sandwich family of chordal graphs. As experimental data often yield distance values which are known to be under-estimated, we address the issue of correcting the data by increasing the distances which are incorrect. This is done by computing, for each graph of the sandwich family, a maximal chordal subgraph.

The size of chordal, interval and threshold subgraphs

Given a graphG withn vertices andm edges, how many edges must be in the largest chordal subgraph ofG? Form=n2/4+1, the answer is 3n/2−1. Form=n2/3, it is 2n−3. Form=n2/3+1, it is at least 7n/3−6 and at most 8n/3−4. Similar questions are studied, with less complete results, for threshold graphs, interval graphs, and the stars on edges, triangles, andK4's.

Maximal chordal subgraphs

An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(| E| Δ), where | E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations.