Lexicographic Breadth First Search Research Articles

Recognizing LBFS trees of bipartite graphs

The graph searches Breadth First Search (BFS) and Depth First Search (DFS) and the spanning trees constructed by them are some of the most basic concepts in algorithmic graph theory. BFS trees are first-in trees, i.e., every vertex is connected to its first visited neighbor. DFS trees are last-in trees, i.e., every vertex is connected to the last visited neighbor before it. The problem whether a given spanning tree can be the first-in tree or last-in tree of a graph search ordering was introduced in the 1980s and has been studied for several graph searches and graph classes. Here, we consider the problem of deciding whether a given spanning tree of a bipartite graph can be a first-in tree or a last-in tree of the Lexicographic Breadth First Search (LBFS), a special variant of BFS that is commonly used in graph algorithms. We show that the recognition of both first-in trees and last-in trees of LBFS is NP-hard even if the start vertex of the search ordering is fixed and the height of the tree is four. We prove that the bound on the height is tight (unless P=NP) by showing that for all spanning trees of bipartite graphs with height smaller than four we can solve both search tree recognition problems of LBFS in polynomial time. Finally, we give a linear-time algorithm that solves both problems for chordal bipartite graphs and fixed start vertices.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

Medians in median graphs and their cube complexes in linear time

The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P. In this paper, we present a linear time algorithm to compute medians in median graphs. We also present a linear time algorithm to compute medians in the associated ℓ1-cube complexes. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges (Θ-classes) via Lexicographic Breadth First Search (LexBFS). We show that any LexBFS ordering of the vertices of a median graph satisfies the following fellow traveler property: the parents of any two adjacent vertices are also adjacent. Using the fast computation of the Θ-classes, we also compute the Wiener index (total distance) in linear time and the distance matrix in optimal quadratic time.

Lexicographic Breadth First-Search and Branch and Bound Algorithms for the Maximum Clique Problem

Lexicographic Breadth First-Search and Branch and Bound Algorithms for the Maximum Clique Problem

Big Data Processing on Volunteer Computing

In order to calculate the node big data contained in complex networks and realize the efficient calculation of complex networks, based on voluntary computing, taking ICE middleware as the communication medium, the loose coupling distributed framework DCBV based on voluntary computing is proposed. Then, the Master, Worker, and MiddleWare layers in the framework, and the development structure of a DCBV framework are designed. The task allocation and recovery strategy, message passing and communication mode, and fault tolerance processing are discussed. Finally, to calculate and verify parameters such as the average shortest path of the framework and shorten calculation time, an improved accurate shortest path algorithm, the N-SPFA algorithm, is proposed. Under different datasets, the node calculation and performance of the N-SPFA algorithm are explored. The algorithm is compared with four approximate shortest-path algorithms: Combined Link and Attribute (CLA), Lexicographic Breadth First Search (LBFS), Approximate algorithm of shortest path length based on center distance of area division (CDZ), and Hub Vertex of area and Core Expressway (HEA-CE). The results show that when the number of CPU threads is 4, the computation time of the DCBV framework is the shortest (514.63 ms). As the number of CPU cores increases, the overall computation time of the framework decreases gradually. For every 2 additional CPU cores, the number of tasks increases by 1. When the number of Worker nodes is 8 and the number of nodes is 1, the computation time of the framework is the shortest (210,979 ms), and the IO statistics data increase with the increase of Worker nodes. When the datasets are Undirected01 and Undirected02, the computation time of the N-SPFA algorithm is the shortest, which is 4520 ms and 7324 ms, respectively. However, the calculation time in the ca-condmat_undirected dataset is 175,292 ms, and the performance is slightly worse. Overall, however, the performance of the N-SPFA and SPFA algorithms is good. Therefore, the two algorithms are combined. For networks with less complexity, the computational scale coefficient of the SPFA algorithm can be set to 0.06, and for general networks, 0.2. When compared with other algorithms in different datasets, the pretreatment time, average query time, and overall query time of N-SPFA algorithm are the shortest, being 49.67 ms, 5.12 ms, and 94,720 ms, respectively. The accuracy (1.0087) and error rate (0.024) are also the best. In conclusion, voluntary computing can be applied to the processing of big data, which has a good reference significance for the distributed analysis of large-scale complex networks.

End-vertices of LBFS of (AT-free) bigraphs

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result of Rose, Tarjan and Lueker on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, including chordal graphs, split graphs, interval graphs, and asteroidal triple-free (AT-free) graphs. In this paper we study the end-vertices of LBFS of bipartite graphs. We show that deciding whether a vertex of a bipartite graph is the end-vertex of an LBFS is an NP-complete problem. In contrast we characterize the end-vertices of LBFS of AT-free bipartite graphs (or equivalently, proper interval bigraphs). Our characterization implies that the problem of deciding whether a vertex of an AT-free bipartite graph is the end-vertex of an LBFS is solvable in polynomial time.

A Lex-BFS-based recognition algorithm for Robinsonian matrices

Robinsonian matrices arise in the classical seriation problem and play an important role in many applications where unsorted similarity (or dissimilarity) information must be reordered. We present a new polynomial time algorithm to recognize Robinsonian matrices based on a new characterization of Robinsonian matrices in terms of straight enumerations of unit interval graphs. The algorithm is simple and is based essentially on lexicographic breadth-first search (Lex-BFS), using a divide-and-conquer strategy. When applied to a nonnegative symmetric n×n matrix with m nonzero entries and given as a weighted adjacency list, it runs in O(d(n+m)) time, where d is the depth of the recursion tree, which is at most the number of distinct nonzero entries of A.

Computing a Clique Tree with the Algorithm Maximal Label Search

The algorithm MLS (Maximal Label Search) is a graph search algorithm that generalizes the algorithms Maximum Cardinality Search (MCS), Lexicographic Breadth-First Search (LexBFS), Lexicographic Depth-First Search (LexDFS) and Maximal Neighborhood Search (MNS). On a chordal graph, MLS computes a PEO (perfect elimination ordering) of the graph. We show how the algorithm MLS can be modified to compute a PMO (perfect moplex ordering), as well as a clique tree and the minimal separators of a chordal graph. We give a necessary and sufficient condition on the labeling structure of MLS for the beginning of a new clique in the clique tree to be detected by a condition on labels. MLS is also used to compute a clique tree of the complement graph, and new cliques in the complement graph can be detected by a condition on labels for any labeling structure. We provide a linear time algorithm computing a PMO and the corresponding generators of the maximal cliques and minimal separators of the complement graph. On a non-chordal graph, the algorithm MLSM, a graph search algorithm computing an MEO and a minimal triangulation of the graph, is used to compute an atom tree of the clique minimal separator decomposition of any graph.

Similarity-First Search: A New Algorithm with Application to Robinsonian Matrix Recognition

We present a new efficient combinatorial algorithm for recognizing if a given symmetric matrix is Robinsonian, i.e., if its rows and columns can be simultaneously reordered so that entries are monotone nondecreasing in rows and columns when moving toward the diagonal. As the main ingredient we introduce a new algorithm, named Similarity-First Search (SFS), which extends lexicographic breadth-first search (Lex-BFS) to weighted graphs and which we use in a multisweep algorithm to recognize Robinsonian matrices. Since Robinsonian binary matrices correspond to unit interval graphs, our algorithm can be seen as a generalization to weighted graphs of the 3-sweep Lex-BFS algorithm of Corneil for recognizing unit interval graphs. This new recognition algorithm is extremely simple and it exploits new insight on the combinatorial structure of Robinsonian matrices. For an $n\times n$ nonnegative matrix with $m$ nonzero entries, it terminates in $n-1$ SFS sweeps, with overall running time $O(n^2 +nm\log n)$.

Efficiently decomposing, recognizing and triangulating hole-free graphs without diamonds

A graph is hole- and diamond-free (HD-free) if none of its induced subgraphs is isomorphic to a chordless cycle of length at least five or to a diamond. Using the clique separator approach and the simple structure of atoms of HD-free graphs, we show how to recognize HD-free graphs in time O(n2). One of the main tools is Lexicographic Breadth-First Search (LexBFS); we give two new properties of LexBFS which are essential for reaching the time bound and which hold for any graph. Moreover, we find minimal triangulations of HD-free graphs in time O(n2), introducing efficient algorithms for the minimal triangulation of matched co-bipartite graphs and chordal bipartite graphs.

Influence of the tie-break rule on the end-vertex problem

End-vertices of a given graph search may have some nice properties, as for example it is well known that the last vertex of Lexicographic Breadth First Search (LBFS) in a chordal graph is simplicial, see Rose, Tarjan and Lueker 1976. Therefore it is interesting to consider if these vertices can be recognized in polynomial time or not, as first studied in Corneil, Köhler and Lanlignel 2010. A graph search is a mechanism for systematically visiting the vertices of a graph. At each step of a graph search, the key point is the choice of the next vertex to be explored. Graph searches only differ by this selection mechanism during which a tie-break rule is used. In this paper we study how the choice of the tie-break in case of equality during the search, for a given graph search including the classic ones such as BFS and DFS, can determine the complexity of the end-vertex problem. In particular we prove a counterintuitive NP-completeness result for Breadth First Search solving a problem raised in Corneil, Köhler and Lanlignel 2010.

A four-sweep LBFS recognition algorithm for interval graphs

Graph Theory In their 2009 paper, Corneil et al. design a linear time interval graph recognition algorithm based on six sweeps of Lexicographic Breadth-First Search (LBFS) and prove its correctness. They believe that their corresponding 5-sweep LBFS interval graph recognition algorithm is also correct. Thanks to the LBFS structure theory established mainly by Corneil et al., we are able to present a 4-sweep LBFS algorithm which determines whether or not the input graph is a unit interval graph or an interval graph. Like the algorithm of Corneil et al., our algorithm does not involve any complicated data structure and can be executed in linear time.

Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time O(n+m)?(n+m), where ? is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear time split decomposition algorithm (Dahlhaus in J. Algorithms 36(2):205---240, 2000), which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper (Gioan et al. in arXiv:1104.3284 , 2011) uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.

Practical and Efficient Circle Graph Recognition

Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n+m) times the inverse Ackermann function, α(n+m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.

Maximal Neighborhood Search and Rigid Interval Graphs

A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic Breadth-First Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the so-called Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several dierent ways. This allows us obtain several linear time multi-sweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3-sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2-sweep MNS certifying recognition algorithm.

The LBFS Structure and Recognition of Interval Graphs

A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple–free graphs, two families where the well-known lexicographic breadth first search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and that by exploiting this structure one can design a linear time, easily implementable, interval graph recognition algorithm.

On end-vertices of Lexicographic Breadth First Searches

Recently Lexicographic Breadth First Search (LBFS) has received considerable attention and has often been employed in a multi-sweep fashion. One variant of LBFS called LBFS+ breaks ties by choosing the last vertex of the tied set in a previous LBFS. This has motivated the study of vertices that may appear last in an LBFS (called end-vertices). In this paper, we present various theoretical and algorithmic results concerning end-vertices.

A Simple Linear Time LexBFS Cograph Recognition Algorithm

Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as $P_4$-reducible, $P_4$-sparse, and distance hereditary. It is an open question whether our cograph recognition algorithm can be extended to a similarly easy algorithm for modular decomposition.

Coloring Bull-Free Perfectly Contractile Graphs

We consider the class of graphs that contain no bull, no odd hole, and no antihole of length at least five. We present a new algorithm that colors optimally the vertices of every graph in this class. This algorithm is based on the existence in every such graph of an ordering of the vertices with a special property. More generally we prove, using a variant of lexicographic breadth-first search, that in every graph that contains no bull and no hole of length at least five there is a vertex that is not the middle of a chordless path on five vertices. This latter fact also generalizes known results about chordal bipartite graphs, totally balanced matrices, and strongly chordal graphs.

A linear time algorithm to list the minimal separators of chordal graphs

Kumar and Madhavan [Minimal vertex separators of chordal graphs, Discrete Appl. Math. 89 (1998) 155–168] gave a linear time algorithm to list all the minimal separators of a chordal graph. In this paper we give another linear time algorithm for the same purpose. While the algorithm of Kumar and Madhavan requires that a specific type of PEO, namely the MCS PEO is computed first, our algorithm works with any PEO. This is interesting when we consider the fact that there are other popular methods such as Lex BFS to compute a PEO for a given chordal graph.