Vertex Elimination Research Articles

First Entire Zagreb Index of Fuzzy Graph and Its Application

The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of α-cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields.

Сравнение алгоритма на основе дескриптора Фурье с алгоритмами, опирающимися на геометрические критерии

Line generalization is an essential data processing operation in geographic information systems and cartography. Many point reduction and simplification algorithms have been developed for this purpose. In 2020 a multi-scale representation model of polyline based on a Fourier descriptor was proposed. The authors present a comparative study of such an algorithm and several simplification ones (Douglas – Peucker, Visvalingham – Whyatt, Li – Openshaw, sleeve-fitting), with different criteria for vertex elimination. A brief description of each method involved in the comparison and a more detailed one for that based on the Fourier descriptor is provided. Generalization quality evaluations characterizing the accuracy of the location and geographical plausibility of the line are discussed. Three coastlines of 1

BPRIA: Crosstalk-Avoided Bi-Partitioning-Based Counter-Propagation Resource Identification and Allocation for Spectrally-Spatially Elastic Optical Networks

Signal transmission using counter-propagation nowadays is adopted to enhance resource utilization in spectrally-spatially elastic optical networks (SS-EONs), where inter-mode and inter-core crosstalks always degrade signal quality and become bottlenecks for high transport capacity. In this paper, we propose BPRIA for the first time, a Bi-Partitioning-based crosstalk-avoided Resource Identification and Allocation scheme in SS-EONs to enhance resource utilization while suppressing both inter-mode and inter-core crosstalks. We introduce a bi-partitioning optimization problem with vertex elimination to maximize the number of non-adjacent cores and modes in each partition of the bipartite graph; two distinct sets of cores and modes are used in a counter-propagation manner for lightpath allocation. The optimization problem is formulated as an integer linear programming (ILP) problem, and we prove that the optimization problem is NP-complete. An algorithm for core-mode-spectrum allocation is developed considering two distinct sets of cores and modes obtained by ILP to serve lightpath requests while avoiding inter-mode and inter-core crosstalks. For dynamic scenarios, we present core-mode-spectrum allocation. Numerical results reveal that the blocking probability is reduced by BPRIA in SS-EONs, and it enhances traffic admissibility in the network.

Propositional Encodings of Acyclicity and Reachability by Using Vertex Elimination

We introduce novel methods for encoding acyclicity and s-t-reachability constraints for propositional formulas with underlying directed graphs, based on vertex elimination graphs, which makes them suitable for cases where the underlying graph has a low directed elimination width. In contrast to solvers with ad hoc constraint propagators for graph constraints such as GraphSAT, our methods encode these constraints as standard propositional clauses, making them directly applicable with any SAT solver. An empirical study demonstrates that our methods do often outperform both earlier encodings of these constraints as well as GraphSAT especially when underlying graphs have a low directed elimination width.

Efficient Computation and Informative Estimation of h+ by Integer and Linear Programming

We investigate modeling cost optimal delete-free STRIPS Planning by Integer/Linear Programming (IP/LP). We introduce two IP models and their LP relaxations based on a recently formulated representation of relaxed plans, named causal relaxed plan representation. The new models are produced by enforcing acyclicity in so-called causal relation graphs using vertex elimination and time labeling methods. We empirically show that while the vertex elimination based method outperforms the time labeling based method and all previously introduced domain independent methods for computing the exact values of h+, the time labeling based LP model is faster to solve compared to its vertex elimination based alternative, making it more suitable for using as heuristic function for optimal planning. We also theoretically analyze the admissible heuristic functions obtained by solving our LP models, and prove that the vertex elimination based heuristic is at least as informative as the time labeling based heuristic. Moreover, our empirical analysis shows that our vertex elimination based heuristic, which is a novel admissible estimation of h+, often has information complementary to that of the LM-cut heuristic.

Whitney's connectivity inequalities for directed hypergraphs

Whitney’s inequality established an important connection between vertex and edge connectivity and the degree of a graph, which was later generalized to digraphs and to undirected hypergraphs. Here we show, using the most common definitions of connectedness for directed hypergraphs, that an analogous result holds directed hypergraphs. It relates the vertex connectivity under strong vertex elimination, edge connectivity under weak edge elimination, and a suitable degree-like parameter and is a proper generalization of the situation in both digraphs and undirected hypergraphs. We furthermore relate the connectivity parameters of directed hypergraphs with those of its directed bipartite König representation.

Linear‐time algorithms for eliminating claws in graphs

AbstractSince many ‐complete graph problems are polynomial‐time solvable when restricted to claw‐free graphs, we study the problem of determining the distance of a given graph to a claw‐free graph, considering vertex elimination a measure. Claw‐free Vertex Deletion (CFVD) consists of determining the minimum number of vertices to be removed from a graph such that the resulting graph is claw‐free. Although CFVD is ‐hard in general and recognizing claw‐free graphs is still a challenge, where the current best deterministic algorithm for a graph G consists of performing executions of the best algorithm for matrix multiplication, we present linear‐time algorithms for CFVD on weighted block graphs and weighted graphs with bounded treewidth. Furthermore, we show that this problem on forests can be solved in linear time by a simpler algorithm, and we determine the exact values for full k‐ary trees. On the other hand, we show that CFVD is ‐hard even when the input graph is a split graph. We also show that the problem is hard to be approximated within any constant factor better than 2, assuming the unique games conjecture.

On lower bounds for optimal Jacobian accumulation

The task of finding a way to compute by using the minimal number of multiplications is generally referred to as the Optimal Jacobian Accumulation () problem. Often a special case of the problem is examined, where the accumulation of the Jacobian is considered as a transformation of the linearized directed acyclic graph (l-DAG) G into , such that is a subgraph of the complete directed bipartite graph . The transformation of the l-DAG is performed by elimination methods. The most commonly used elimination methods are vertex elimination and edge elimination. The problem of transforming an l-DAG into a bipartite graph by vertex or edge eliminations with minimal costs is generally referred as the Vertex Elimination () or Edge Elimination () problem. Both the and problems are conjectured to be NP-hard. Thus Branch and bound based algorithms in addition to greedy heuristics are used to solve these problems. For efficient application of such algorithms, good lower bounds are essential. In this paper, we develop a new approach for computing the lower bounds for the optimal solution of the and problems.

Perfect elimination orderings for symmetric matrices

We introduce a new class of structured symmetric matrices by extending the notion of perfect elimination ordering from graphs to weighted graphs or matrices. This offers a common framework capturing common vertex elimination orderings of monotone families of chordal graphs, Robinsonian matrices and ultrametrics. We give a structural characterization for matrices that admit perfect elimination orderings in terms of forbidden substructures generalizing chordless cycles in graphs.

Characterizations and algorithms for generalized Cops and Robbers games

We propose a definition of generalized Cops and Robbers games where there are two players, the Pursuer and the Evader, who each move via prescribed rules. If the Pursuer can ensure that the gameenters into a fixed set of final positions, then the Pursuer wins; otherwise, the Evader wins. A relational characterization of the games where the Pursuer wins is provided. A precise formula isgiven for the length of the game, along with an algorithm for computing if the Pursuer has a winning strategy whose complexity is a function of the parameters of the game. For games where theposition of one player does not affect the available moves of he other, a vertex elimination ordering characterization, analogous to a cop-win ordering, is given for when the Pursuer has a winningstrategy.

Vertex elimination orderings for hereditary graph classes

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vušković.

Elimination schemes and lattices

Perfect vertex elimination schemes are part of the characterizations for several classes of graphs, including chordal and cop-win. Partial elimination schemes reduce a graph to an important subgraph, for example, k-cores and robber-win graphs. We are interested in those partial elimination schemes, in which once a vertex is ready to be eliminated, it stays in that state regardless of which other vertices are eliminated. We show that in such a scheme, the sets of subsets of eliminated vertices, when ordered by inclusion, form an upper locally distributed lattice. We also show that (a) unless they contain a specific induced subgraph, the cop-win orderings have this property, and that (b) the process of cleaning graphs also leads to upper locally distributed lattices. Finally, we ask for an elimination scheme, which graphs are associated with distributive lattices?

Vertex Ordering Characterizations of Graphs of Bounded Asteroidal Number

Asteroidal Triple-free (AT-free) graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component. (AT-free graphs have asteroidal number at most 2.) In this article, we characterize graphs of bounded asteroidal number by means of a vertex elimination ordering, thereby solving a long-standing open question in algorithmic graph theory. Similar characterizations are known for chordal, interval, and cocomparability graphs.

Modifying a Graph Using Vertex Elimination

Vertex elimination is a graph operation that turns the neighborhood of a vertex into a clique and removes the vertex itself. It has widely known applications within sparse matrix computations. We define the Elimination problem as follows: given two graphs G and H, decide whether H can be obtained from G by |V(G)|?|V(H)| vertex eliminations. We show that Elimination is $\mathsf {W[1]} $ -hard when parameterized by |V(H)|, even if both input graphs are split graphs, and $\mathsf {W[2]} $ -hard when parameterized by |V(G)|?|V(H)|, even if H is a complete graph. On the positive side, we show that Elimination admits a kernel with at most 5|V(H)| vertices in the case when G is connected and H is a complete graph, which is in sharp contrast to the $\mathsf {W[1]} $ -hardness of the related Clique problem. We also study the case when either G or H is tree. The computational complexity of the problem depends on which graph is assumed to be a tree: we show that Elimination can be solved in polynomial time when H is a tree, whereas it remains NP-complete when G is a tree.

Digraphs of bounded elimination width

Hunter and Kreutzer recently introduced the digraph width measure Kelly-width. The measure is based on a vertex elimination process that aims at bounding the out-degree of a vertex during the elimination. We refine the Kelly-width measure by bounding both the out-degree and the in-degree of a vertex. We show that the elimination process and a subgraph characterisation of Kelly-width naturally generalise to the refined notion. The main result of the paper is a game characterisation of the refined measure, which is a surprisingly non-trivial generalisation of the game for Kelly-width.

Fast local search for the steiner problem in graphs

We present efficient algorithms that implement four local searches for the Steiner problem in graphs: vertex insertion, vertex elimination, key-path exchange, and key-vertex elimination. In each case, we show how to find an improving solution (or prove that none exists in the neighborhood) in <it>O</it>(<it>m</it> log <it>n</it>) time on graphs with <it>n</it> vertices and <it>m</it> edges. Many of the techniques and data structures we use are relevant in the study of dynamic graphs in general, beyond Steiner trees. Besides the theoretical interest, our results have practical impact: These local searches have been shown to find good-quality solutions in practice, but high running times limited their applicability.

Computational complexity of some intelligent computing systems

PurposeAs agent‐based systems are increasingly used to model real‐life applications such as the internet, electronic markets or disaster management scenarios, it is important to study the computational complexity of such usually combinatorial systems with respect to some desirable properties. The purpose of this paper is to consider two computational models: graphical games encoding the interactions between rational and selfish agents; and weighted directed acyclic graphs (DAG) for evaluating derivatives of numerical functions. The author studies the complexity of a certain number of search problems in both models.Design/methodology/approachThe author's approach is essentially theoretical, studying the problem of verifying game‐theoretic properties for graphical games representing interactions between self‐motivated and rational agents, as well as the problem of searching for an optimal elimination ordering in a weighted DAG for evaluating derivatives of functions represented by computer programs.FindingsA certain class of games has been identified for which Nash or Bayesian Nash equilibria can be verified in polynomial time; then, it has been shown that verifying a dominant strategy equilibrium is non‐deterministic polynomial (NP)‐complete even for normal form games. Finally, it has been shown that the optimal vertex elimination ordering for weighted DAGs is NP‐complete.Originality/valueThis paper presents a general framework for graphical games. The presented results are novel and illustrate how modeling real‐life scenarios involving intelligent agents can lead to computationally hard problems while showing interesting cases that are tractable.

Finding a Sun in Building-Free Graphs

Deciding whether an arbitrary graph contains a sun was recently shown to be NP-complete (Hoang in SIAM J Discret Math 23:2156–2162, 2010). We show that whether a building-free graph contains a sun can be decided in O(min{mn 3, m 1.5 n 2}) time and, if a sun exists, it can be found in the same time bound. The class of building-free graphs contains many interesting classes of perfect graphs such as Meyniel graphs which, in turn, contains classes such as hhd-free graphs, i-triangulated graphs, and parity graphs. Moreover, there are imperfect graphs that are building-free. The class of building-free graphs generalizes several classes of graphs for which an efficient test for the presence of a sun is known. We also present a vertex elimination scheme for the class of (building, gem)-free graphs. The class of (building, gem)-free graphs is a generalization of the class of distance hereditary graphs and a restriction of the class of (building, sun)-free graphs.

Vertex-ordering Algorithms for Automatic Differentiation of Computer Codes

In the context of Automatic Differentiation (AD) of functions represented by computer code via the vertex elimination approach first advocated by Griewank and Reese (On the Calculation of Jacobian Matrices by the Markowitz Rule. In Griewank, A. and Corliss, G.F. (eds), Automatic Differentiation of Algorithms: Theory, Implementation and Application, pp. 126–135. SIAM, 1991, Philadelphia, PA.), we present two approximate algorithms based on the linearized computational graph of the input code. The first is a statement-reordering algorithm aiming to tune the AD-generated code so as to maximize its performance for modern superscalar processors. The second is aimed at detecting interface contractions introduced by Bischof and Haghighat (Hierarchical Approaches to Automatic Differentiation. In Berz, M., Bischof, C., Corliss, G. and Griewank, A. (eds), Computational Differentiation: Techniques, Applications, and Tools, pp. 83–94. SIAM, 1996, Philadelphia, PA) in order to enable exploitation of the structure of the input code in the differentiation process. Performance data are also presented.

Fast Automatic Differentiation Jacobians by Compact LU Factorization

For a vector function coded without branches or loops, a code for the Jacobian is generated by interpreting Griewank and Reese's vertex elimination as Gaussian elimination and implementing this as compact LU factorization. Tests on several platforms show such a code is typically 4 to 20 times faster than that produced by tools such as Adifor, Tamc, or Tapenade, on average significantly faster than vertex elimination code produced by the EliAD tool [Tadjouddine et al., in Proceedings of ICCS (2), Lecture Notes in Comput. Sci. 2330, Springer, New York, 2002] and can outperform a hand-coded Jacobian. The LU approach is promising, e.g., for CFD flux functions that are central to assembling Jacobians in finite element or finite volume calculations and, in general, for any inner-loop basic block whose Jacobian is crucial to an overall computation involving derivatives.