Graph Decomposition Research Articles

Decomposition of product graphs into sunlet graphs of order eight

For any integer $k\geq 3$ , we define sunlet graph of order $2k$, denoted by $L_{2k}$, as the graph consisting of a cycle of length $k$ together with $k$ pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for the existence of $L_{8}$-decomposition of tensor product and wreath product of complete graphs.

Counting Weighted Independent Sets beyond the Permanent

Jerrum, Sinclair and Vigoda (2004) showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. Equivalently, this may be viewed as approximating the partition function of vertex-weighted independent sets in the line graph of a bipartite graph. Line graphs of bipartite graphs are perfect graphs, and are known to be precisely the class of (claw, diamond, odd hole)-free graphs. So how far does the result of Jerrum, Sinclair and Vigoda extend? We first show that it extends to (claw, odd hole)-free graphs, and then show that it extends to the even larger class of (fork, odd hole)-free graphs. Our techniques are based on graph decompositions, which have been the focus of much recent work in structural graph theory, and on structural results of Chvatal and Sbihi (1988), Maffray and Reed (1999) and Lozin and Milanic (2008).

Irregularity Strength of Circulant Graphs Using Algorithmic Approach

This paper deals with decomposition of complete graphs on $n$ vertices into circulant graphs with reduced degree $r . They are denoted as $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ , where $a_{1}$ to $a_{m}$ are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why after decomposition, an edge irregular $k$ -labeling for these subgraphs is computed with the help of algorithmic approach. Results of $k$ are computed by implementing this iterative algorithm in computer. Using the values of $k$ , an upper bound for edge irregularity strength is suggested for $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ that is ${\vert E\vert }/{2}\log _{2} \vert V\vert $ .

Decomposition of wheel graphs into stars, cycles and paths

Decomposition of wheel graphs into stars, cycles and paths

Decompositions of complete bipartite graphs and complete graphs into paths, stars, and cycles with four edges each

Decompositions of complete bipartite graphs and complete graphs into paths, stars, and cycles with four edges each

Containment of rumor spread by selecting immune nodes in social networks.

With the popularity of online social network these have become important platforms for the spread of information. This not only includes correct and useful information, but also false information, and even rumors which could result in panic. Therefore, the containment of rumor spread in social networks is important. In this paper, we propose an effective method that involves selecting a set of nodes in (k, η)-cores and immunize these nodes for rumor containment. First, we study rumor influence propagation in social networks under the extended Independent Cascade (EIC) model, an extension of the classic Independent Cascade (IC) model. Then, we decompose a social network into subgraphs via core decomposition of uncertain graphs and compute the number of immune nodes in each subgraph. Further we greedily select nodes with the Maximum Marginal Covering Neighbors Set as immune nodes. Finally, we conduct experiments using real-world datasets to evaluate our method. Experimental results show the effectiveness of our method.

SyDeBO: Symbolic-Decision-Embedded Bilevel Optimization for Long-Horizon Manipulation in Dynamic Environments

This study proposes a Task and Motion Planning (TAMP) method with symbolic decisions embedded in a bilevel optimization. This TAMP method exploits the discrete structure of sequential manipulation for long-horizon and versatile tasks in dynamic environments. At the symbolic planning level, we propose a scalable decision-making method for long-horizon manipulation tasks using the Planning Domain Definition Language (PDDL) with causal graph decomposition. At the motion planning level, we devise a trajectory optimization (TO) approach based on the Differential Dynamic Programming (DDP) and Alternating Direction Method of Multipliers (ADMM), suitable for solving constrained, large-scale nonlinear optimization in a distributed manner. Distinct from conventional geometric motion planners, our approach generates highly dynamic manipulation motions by incorporating the full robot and object dynamics. Furthermore, in lieu of a hierarchical planning approach, we solve a holistically integrated bilevel optimization problem involving costs from both the low-level TO and the high-level search. Simulation and experimental results demonstrate dynamic manipulation for long-horizon object sorting tasks in clutter and on a moving conveyor belt.

Human Interaction Understanding With Joint Graph Decomposition and Node Labeling.

The task of human interaction understanding involves both recognizing the action of each individual in the scene and decoding the interaction relationship among people, which is useful to a series of vision applications such as camera surveillance, video-based sports analysis and event retrieval. This paper divides the task into two problems including grouping people into clusters and assigning labels to each of them, and presents an approach to solving these problems in a joint manner. Our method does not assume the number of groups is known beforehand as this will substantially restrict its application. With the observation that the two challenges are highly correlated, the key idea is to model the pairwise interacting relations among people via a complete graph and its associated energy function such that the labeling and grouping problems are translated into the minimization of the energy function. We implement this joint framework by fusing both deep features and rich contextual cues, and learn the fusion parameters from data. An alternating search algorithm is developed in order to efficiently solve the associated inference problem. By combining the grouping and labeling results obtained with our method, we are able to achieve the semantic-level understanding of human interactions. Extensive experiments are performed to qualitatively and quantitatively evaluate the effectiveness of our approach, which outperforms state-of-the-art methods on several important benchmarks. An ablation study is also performed to verify the effectiveness of different modules within our approach.

A Decomposition of a Complete Graph with a Hole

In the field of design theory, the most well-known design is a Steiner Triple System. In general, a G-design on H is an edge-disjoint decomposition of H into isomorphic copies of G. In a Steiner Triple system, a complete graph is decomposed into triangles. In this paper we let H be a complete graph with a hole and G be a complete graph on four vertices minus one edge, also referred to as a . A complete graph with a hole, , consists of a complete graph on d vertices, , and a set of independent vertices of size v, V, where each vertex in V is adjacent to each vertex in . When d is even, we give two constructions for the decomposition of a complete graph with a hole into copies of : the Alpha-Delta Construction, and the Alpha-Beta-Delta Construction. By restricting d and v so that , we are able to resolve both of these cases for a subset of using difference methods and 1-factors.

On new algorithmic techniques for the weighted vertex coloring problem

The classical NP-hard weighted vertex coloring problem consists in minimizing the number of colors in colorings of vertices of a given graph so that, for each vertex, the number of its colors equals a given weight of the vertex and adjacent vertices receive distinct colors. The weighted chromatic number is the smallest number of colors in these colorings. There are several polynomial-time algorithmic techniques for designing efficient algorithms for the weighted vertex coloring problem. For example, standard techniques of this kind are the modular graph decomposition and the graph decomposition by separating cliques. This article proposes new polynomial-time methods for graph reduction in the form of removing redundant vertices and recomputing weights of the remaining vertices so that the weighted chromatic number changes in a controlled manner. We also present a method of reducing the weighted vertex coloring problem to its unweighted version and its application. This paper contributes to the algorithmic graph theory.

Decomposition of complete tripartite graphs into cycles and paths of length three

Let $C_{k}$ and $P_{k}$ denote a cycle and a path on $k$ vertices, respectively. In this paper, we obtain necessary and sufficient conditions for the decomposition of $K_{{r},{s},{t}}$ into $p$ copies of $C_{3}$ and $q$ copies of $P_{4}$ for all possible values of $p$, $q\geq0$.

A complete solution to the spectrum problem for graphs with six vertices and up to nine edges

Let $G$ be a graph. A $G$-design of order $n$ is a decomposition of the complete graph $K_n$ into disjoint copies of $G$. The existence problem of graph designs has been completely solved for all graphs with up to five vertices, and all graphs with six vertices and up to seven edges; and almost completely solved for all graphs with six vertices and eight edges leaving two cases of order 32 unsettled. Up to isomorphism there are 20 graphs with six vertices and nine edges (and no isolated vertex). The spectrum problem has been solved completely for 11 of these graphs, and partially for 2 of these graphs. In this article, the two missing graph designs for the six-vertex eight-edge graphs are constructed, and a complete solution to the spectrum problem for the six-vertex nine-edge graphs is given; completing the spectrum problem for all graphs with six vertices and up to nine edges.

Fault Diagnosis by Multisensor Data: A Data-Driven Approach Based on Spectral Clustering and Pairwise Constraints.

This paper deals with clustering based on feature selection of multisensor data in high-dimensional space. Spectral clustering algorithms are efficient tools in signal processing for grouping datasets sampled by multisensor systems for fault diagnosis. The effectiveness of spectral clustering stems from constructing an embedding space based on an affinity matrix. This matrix shows the pairwise similarity of the data points. Clustering is then obtained by determining the spectral decomposition of the Laplacian graph. In the manufacturing field, clustering is an essential strategy for fault diagnosis. In this study, an enhanced spectral clustering approach is presented, which is augmented with pairwise constraints, and that results in efficient identification of fault scenarios. The effectiveness of the proposed approach is described using a real case study about a diesel injection control system for fault detection.

Multivariate statistical matching using graphical modeling

The goal of statistical matching, at a macro level, is the estimation of the joint distribution of variables separately observed in independent samples. The lack of joint information on the variables of interest leads to uncertainty about the data generating model. In this paper we propose the use of graphical models to deal with the statistical matching uncertainty for multivariate categorical variables. The use of Bayesian networks in the statistical matching context allows both to introduce extra sample information on the dependence structure between the variables of interest and to use such an information to factorize the joint probability distribution according to the graph decomposition of a multivariate dependence in lower dimension components. This representation of the joint probability distribution, taking advantage of local relationships, allows to simplify both parameters estimation and statistical matching quality evaluation in a multivariate context. A simulation experiment is performed in order to evaluate the performance of the proposed methodology with and without auxiliary information, as well as to compare it with the saturated multinomial model, in terms of uncertainty reduction. Finally, an application to a real case is provided. Results show a considerable improvement in the quality of statistical matching when the dependence structure is taken into account.

Graph decompositions in projective geometries

AbstractLet PG be the ‐dimensional projective space over and let be a simple graph of order for some . A design over is a collection of graphs (blocks) isomorphic to with the following properties: the vertex set of every block is a subspace of PG; every two distinct points of PG are adjacent in exactly blocks. This new definition covers, in particular, the well‐known concept of a design over corresponding to the case that is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of ‐decompositions over or for which is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that is complete and , that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable designs over . This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set.

On the super (a, d)-H-antimagic total labelings of path chain graphs

Super (a, d)-H-antimagic total labeling is closely related to graph labeling, graph decomposition, graph covering and magic square and it arouses more and more people’s concerns. In this paper we study the super (a, d)-H-antimagic total labelings of path chain graph PCG(H,U, n), when d = 0, 1, 2, 3, 4.

Decompositions of 6-Regular Bipartite Graphs into Paths of Length Six

Let T be a tree with m edges. It was conjectured that every m-regular bipartite graph can be decomposed into edge-disjoint copies of T. In this paper, we prove that every 6-regular bipartite graph can be decomposed into edge-disjoint paths with 6 edges. As a consequence, every 6-regular bipartite graph on n vertices can be decomposed into $$\frac{n}{2}$$ paths, which is related to the well-known Gallai’s Conjecture: every connected graph on n vertices can be decomposed into at most $$\frac{n+1}{2}$$ paths.

On the star decomposition of a graph: Hardness results and approximation for the max–min optimization problem

We study the problem of decomposing a graph into stars so that the minimum size star in the decomposition is as large as possible. Problems of graph decomposition have been actively investigated since the 70’s. The question we consider here also combines the structure of a facility location problem (choosing the centres of the stars) with a max–min fairness optimization criterion that has recently received attention in resource allocation problems, e.g., the Santa Claus problem.We focus on computational and algorithmic questions: We show that the problem is hard even in the case of planar graphs of maximum degree not larger than four, and already for decompositions into stars of size at least three. We are able to tightly characterize the boundaries between efficiently solvable instances and hard ones: we show that relaxing any of the conditions in our hardness result (minimum size of the stars or degree of the input graph) makes the problem polynomially solvable.Our complexity result implies also the APX hardness of the problem ruling out any approximation guarantee better than 2/3. We complement this inapproximability result with a 1/2-approximation algorithm. Finally, we give a polynomial time algorithm for trees. A nice property of our algorithms is that they can all be implemented to run in time linear in the size of the input graph.

(Di)graph Decompositions and Magic Type Labelings: A Dual Relation

A graph G is called edge-magic if there is a bijective function f from the set of vertices and edges to the set $$\{1,2,\ldots ,|V(G)|+|E(G)|\}$$ such that the sum $$f(x)+f(xy)+f(y)$$ for any xy in E(G) is constant. Such a function is called an edge-magic labeling of G, and the constant is called the valence of f. An edge-magic labeling with the extra property that $$f(V(G))= \{1,2,\ldots ,|V(G)|\}$$ is called super edge-magic. In this paper, we establish a relationship between the valences of (super) edge-magic labelings of certain types of bipartite graphs and the existence of a particular type of decompositions of such graphs.

Alpha graphs with different pendent paths

Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.