Edge Deletion Research Articles

Maintaining centdians in a fully dynamic forest with top trees

In this paper, we consider the problem of maintaining the centdians in a fully dynamic forest. A forest is said to be fully dynamic if edge insertions, edge deletions, and changes of vertex weights are allowed. Centdian is a specific kind of facility that integrates the notions of center and median by taking a convex combination on the objective functions of both problems. This work extends the results in Alstrup et al. [2] within the same time complexity, i.e., linear time preprocessing and O(logn) per update, where n is the number of vertices of the components being updated.

Incompressibility of $$H$$ H -Free Edge Modification Problems

Given a fixed graph $$H$$H, the $$H$$H-Free Edge Deletion (resp., Completion, Editing) problem asks whether it is possible to delete from (resp., add to, delete from or add to) the input graph at most $$k$$k edges so that the resulting graph is $$H$$H-free, i.e., contains no induced subgraph isomorphic to $$H$$H. These $$H$$H-free edge modification problems are well known to be fixed-parameter tractable for every fixed $$H$$H. In this paper we study the incompressibility, i.e., nonexistence of polynomial kernels, for these $$H$$H-free edge modification problems in terms of the structure of $$H$$H, and completely characterize their nonexistence for $$H$$H being paths, cycles or 3-connected graphs. We also give a sufficient condition for the nonexistence of polynomial kernels for $${\mathcal {F}}$$F-Free Edge Deletion problems, where $${\mathcal {F}}$$F is a finite set of forbidden induced subgraphs. As an effective tool, we have introduced an incompressible constraint satisfiability problem Propagational-$$f$$f Satisfiability to express common propagational behaviors of events, and we expect the problem to be useful in studying the nonexistence of polynomial kernels in general.

Duplication and Divergence Effect on Network Motifs in Undirected Bio-Molecular Networks.

Duplication and divergence are two basic evolutionary mechanisms of bio-molecular networks. Real-world bio-molecular networks and their statistical characteristics can be well mimicked by artificial algorithms based on the two mechanisms. Bio-molecular networks consist of network motifs, which act as building blocks of large-scale networks. A fundamental question is how network motifs are evolved from long time evolution and natural selection. By considering the effect of various duplication and divergence strategies, we find that the underlying duplication scheme of the real-world undirected bio-molecular networks would rather follow the anti-preference strategy than the random one. The anti-preference duplication mechanism and the dimerization processes can lead to the formation of various motifs, and robustly conserve proper quantities of motifs in the artificial networks as that in the real-world ones. Furthermore, the anti-preference mechanism and edge deletion divergence can robustly preserve the sparsity of the networks. The investigations reveal the possible evolutionary mechanisms of network motifs in real-world bio-molecular networks, and have potential implications in the design, synthesis and reengineering of biological networks for biomedical purpose.

The sequential equal surplus division for rooted forest games and an application to sharing a river with bifurcations

We introduce a new allocation rule, called the sequential equal surplus division for rooted forest TU-games. We provide two axiomatic characterizations for this allocation rule. The first one uses the classical property of component efficiency plus an edge deletion property. The second characterization uses standardness, an edge deletion property applied to specific rooted trees, a consistency property, and an amalgamation property. We also provide an extension of the sequential equal surplus division applied to the problem of sharing a river with bifurcations.

Graph editing to a fixed target

For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions. Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem. For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H. Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H.

An overview of kernelization algorithms for graph modification problems

Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

Diameter variability of hypercubes

A parallel and distributed system is usually represented by a graph. The maximum communication delay between any pair of processors in a parallel and distributed system can be determined by the diameter of its underlying graph. The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we show that the diameter of an n-dimensional hypercube can be decreased by k with the addition of 22k−1 edges for 1≤k≤⌊n/2⌋.

Efficient and Dynamic Algorithms for Alternating Büchi Games and Maximal End-Component Decomposition

The computation of the winning set for Büchi objectives in alternating games on graphs is a central problem in computer-aided verification with a large number of applications. The long-standing best known upper bound for solving the problem is Õ ( n · m ), where n is the number of vertices and m is the number of edges in the graph. We are the first to break the Õ ( n · m ) boundary by presenting a new technique that reduces the running time to O ( n 2 ). This bound also leads to O ( n 2 )-time algorithms for computing the set of almost-sure winning vertices for Büchi objectives (1) in alternating games with probabilistic transitions (improving an earlier bound of Õ ( n · m )), (2) in concurrent graph games with constant actions (improving an earlier bound of O ( n 3 )), and (3) in Markov decision processes (improving for m > n 4/3 an earlier bound of O ( m · √ m )). We then show how to maintain the winning set for Büchi objectives in alternating games under a sequence of edge insertions or a sequence of edge deletions in O ( n ) amortized time per operation. Our algorithms are the first dynamic algorithms for this problem. We then consider another core graph theoretic problem in verification of probabilistic systems, namely computing the maximal end-component decomposition of a graph. We present two improved static algorithms for the maximal end-component decomposition problem. Our first algorithm is an O ( m · √ m )-time algorithm, and our second algorithm is an O ( n 2 )-time algorithm which is obtained using the same technique as for alternating Büchi games. Thus, we obtain an O (min {m · √ m , n 2 })-time algorithm improving the long-standing O ( n · m ) time bound. Finally, we show how to maintain the maximal end-component decomposition of a graph under a sequence of edge insertions or a sequence of edge deletions in O ( n ) amortized time per edge deletion, and O ( m ) worst-case time per edge insertion. Again, our algorithms are the first dynamic algorithms for this problem.

한 사이클 내에서 최대 가중치 간선을 제거하기 위한 최소 신장트리 알고리즘

Abstract This paper suggests a method that obtains the minimum spanning tree (MST) far more easily and rapidly than the present ones. The suggested algorithm, firstly, simplifies a graph by means of reducing the number of edges of the graph. To achieve this, it applies a method of eliminating the maximum weight edge if the valency of vertices of the graph is equal to or more than 3. As a result of this step, we can obtain the reduced edge population. Next, it applies a method in which the maximum weight edge is eliminated within the cycle. On applying the suggested population minimizing and maximum weight edge deletion algorithms to 9 various graphs, as many as the number of cycles of the graph is executed and MST is easily obtained. It turns out to lessen 66% of the number of cycles and obtain the MST in at least 2 and at most 8 cycles by only deleting the maximum weight edges. Key Words : Minimum Spanning Tree, Valency, Cycle, Maximum Weight Edge Ⅰ. 서 론 그래프  는 정점들 (Vertices, )과 간선들(Edges, )로 구성되어 있으며, 일반적으로 그래프란 모든 정점들을 연결하는 간선들이 무방향성 (Undirected)을 갖고 있는 무방향 그래프 (Undirected Graph)를 의미

Network completion for static gene expression data.

We tackle the problem of completing and inferring genetic networks under stationary conditions from static data, where network completion is to make the minimum amount of modifications to an initial network so that the completed network is most consistent with the expression data in which addition of edges and deletion of edges are basic modification operations. For this problem, we present a new method for network completion using dynamic programming and least-squares fitting. This method can find an optimal solution in polynomial time if the maximum indegree of the network is bounded by a constant. We evaluate the effectiveness of our method through computational experiments using synthetic data. Furthermore, we demonstrate that our proposed method can distinguish the differences between two types of genetic networks under stationary conditions from lung cancer and normal gene expression data.

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

Nonlinear dynamic evolution and control in a new scale-free networks modeling

The nonlinear evolving and controlling in complex networks are an important way to understand the dynamic mechanism for real networks. In order to explore universality of scale-free systems, we propose an extended network model based on Barabasi–Albert model by developing and decaying networks. The novel network evolves by growing and optimizing processes, such as the addition of new nodes and edges, or deletion of edges at every time step. Meanwhile, in order to describe more realistic phenomena of reality, we introduce the fitness to reflect the competition and local event of inner anti-preferential mechanism to delete the edges. We calculate analytically the degree distribution and find that the Barabasi–Albert model is only one of its special cases and the model self-organizes into scale-free networks, moreover, the numerical simulations are in good agreement with the analytical conclusions. The results imply that this extended model has more comprehensive and universal simulation and reflection in complex network topology characters and evolution with practices and applications.

Editing Simple Graphs

We study the complexity of turning a given graph, by edge editing, into a target graph whose critical-clique graph is any fixed graph. The problem came up in practice, in an effort of mining huge word similarity graphs for well structured word clusters. It also adds to the rich field of graph modification problems. We show in a generic way that several variants of this problem are in SUBEPT. As a special case, we give a tight time bound for edge deletion to obtain a single clique and isolated vertices, and we round up this study with NP-completeness results for a number of target graphs.

Covering tree with stars

We study the tree edit distance (TED) problem with edge deletions and edge insertions as edit operations. We reformulate a special case of this problem as Covering Tree with Stars (CTS): given a tree T and a set $$\mathcal {S}$$ S of stars, can we connect the stars in $$\mathcal {S}$$ S by adding edges between them such that the resulting tree is isomorphic to T? We prove that in the general setting, CST is NP-complete, which implies that the TED considered here is also NP-hard, even when both input trees have diameters bounded by 10. We also show that, when the number of distinct stars is bounded by a constant k, CTS can be solved in polynomial time, by presenting a dynamic programming algorithm running in $$O(|V(T)|^2\cdot k\cdot |V(\mathcal{S})|^{2k})$$ O ( | V ( T ) | 2 · k · | V ( S ) | 2 k ) time.

A Bijection for Tricellular Maps

We give a bijective proof for a relation between unicellular, bicellular, and tricellular maps. These maps represent cell complexes of orientable surfaces having one, two, or three boundary components. The relation can formally be obtained using matrix theory (Dyson, 1949) employing the Schwinger-Dyson equation (Schwinger, 1951). In this paper we present a bijective proof of the corresponding coefficient equation. Our result is a bijection that transforms a unicellular map of genus g into unicellular, bicellular or tricellular maps of strictly lower genera. The bijection employs edge cutting, edge contraction, and edge deletion.

BiCluE - Exact and heuristic algorithms for weighted bi-cluster editing of biomedical data

BackgroundThe explosion of biological data has dramatically reformed today's biology research. The biggest challenge to biologists and bioinformaticians is the integration and analysis of large quantity of data to provide meaningful insights. One major problem is the combined analysis of data from different types. Bi-cluster editing, as a special case of clustering, which partitions two different types of data simultaneously, might be used for several biomedical scenarios. However, the underlying algorithmic problem is NP-hard.ResultsHere we contribute with BiCluE, a software package designed to solve the weighted bi-cluster editing problem. It implements (1) an exact algorithm based on fixed-parameter tractability and (2) a polynomial-time greedy heuristics based on solving the hardest part, edge deletions, first. We evaluated its performance on artificial graphs. Afterwards we exemplarily applied our implementation on real world biomedical data, GWAS data in this case. BiCluE generally works on any kind of data types that can be modeled as (weighted or unweighted) bipartite graphs.ConclusionsTo our knowledge, this is the first software package solving the weighted bi-cluster editing problem. BiCluE as well as the supplementary results are available online at http://biclue.mpi-inf.mpg.de.

Uncovering the role of elementary processes in network evolution

The growth and evolution of networks has elicited considerable interest from the scientific community and a number of mechanistic models have been proposed to explain their observed degree distributions. Various microscopic processes have been incorporated in these models, among them, node and edge addition, vertex fitness and the deletion of nodes and edges. The existing models, however, focus on specific combinations of these processes and parameterize them in a way that makes it difficult to elucidate the role of the individual elementary mechanisms. We therefore formulated and solved a model that incorporates the minimal processes governing network evolution. Some contribute to growth such as the formation of connections between existing pair of vertices, while others capture deletion; the removal of a node with its corresponding edges, or the removal of an edge between a pair of vertices. We distinguish between these elementary mechanisms, identifying their specific role on network evolution.

Faster Parameterized Algorithms for Deletion to Split Graphs

An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We initiate a systematic study of parameterized complexity of the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split. We give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely, In addition, we note that our algorithm for Split Edge Deletion adds to the small number of subexponential parameterized algorithms not obtained through bidimensionality (Demaine et al. in J. ACM 52(6): 866---893, 2005), and on general graphs.

A note on contracting claw-free graphs

Discrete Algorithms A graph containment problem is to decide whether one graph called the host graph can be modified into some other graph called the target graph by using a number of specified graph operations. We consider edge deletions, edge contractions, vertex deletions and vertex dissolutions as possible graph operations permitted. By allowing any combination of these four operations we capture the following problems: testing on (induced) minors, (induced) topological minors, (induced) subgraphs, (induced) spanning subgraphs, dissolutions and contractions. We show that these problems stay NP-complete even when the host and target belong to the class of line graphs, which form a subclass of the class of claw-free graphs, i.e., graphs with no induced 4-vertex star. A natural question is to study the computational complexity of these problems if the target graph is assumed to be fixed. We show that these problems may become computationally easier when the host graphs are restricted to be claw-free. In particular we consider the problems that are to test whether a given host graph contains a fixed target graph as a contraction.

Transition of the Degree Sequence in the Random Graph Model of Cooper, Frieze, and Vera

We study the transition of the expected degree sequence from power law to exponential decay in the random graph process introduced by Cooper, Frieze, and Vera. We prove that there is a threshold on the probabilities of introduction of vertices (with probability π1), edges (π2) and deletion of edges (π3) or vertices (π4). If π1 + 2π2 − 2π3 − π4 > 0, the expected fraction of vertices of degree k follows a power law, whereas for π1 + 2π2 − 2π3 − π4 < 0, it decays exponentially fast. This work extends the previous results of Wu et al. and Deijfen and Lindholm to the whole model defined by Cooper et al.