Graph Decomposition Research Articles

Transit index by means of graph decomposition

Transit index by means of graph decomposition

On the Complexity of Computing Treebreadth

During the last decade, metric properties of the bags of tree decompositions of graphs have been studied. Roughly, the length and the breadth of a tree decomposition are the maximum diameter and radius of its bags respectively. The treelength and the treebreadth of a graph are the minimum length and breadth of its tree decompositions respectively. Pathlength and pathbreadth are defined similarly for path decompositions. In this paper, we answer open questions of [Dragan and Kohler, Algorithmica 2014] and [Dragan, Kohler and Leitert, SWAT 2014] about the computational complexity of treebreadth, pathbreadth and pathlength. Namely, we prove that computing these graph invariants is NP-hard. We further investigate graphs with treebreadth one, i.e., graphs that admit a tree decomposition where each bag has a dominating vertex. We show that it is NP-complete to decide whether a graph belongs to this class. We then prove some structural properties of such graphs which allows us to design polynomial-time algorithms to decide whether a bipartite graph, resp., a planar graph (or more generally, a triangle-free graph, resp., a K 3,3-minor-free graph), has treebreadth one.

Fractional Decompositions and the Smallest-eigenvalue Separation

A new method is introduced for bounding the separation between the value of $-k$ and the smallest eigenvalue of a non-bipartite $k$-regular graph. The method is based on fractional decompositions of graphs. As a consequence we obtain a very short proof of a generalization and strengthening of a recent result of Qiao, Jing, and Koolen [Electronic J. Combin. 26(2) (2019), #P2.41] about the smallest eigenvalue of non-bipartite distance-regular graphs.

Recovering rearranged cancer chromosomes from karyotype graphs

BackgroundMany cancer genomes are extensively rearranged with highly aberrant chromosomal karyotypes. Structural and copy number variations in cancer genomes can be determined via abnormal mapping of sequenced reads to the reference genome. Recently it became possible to reconcile both of these types of large-scale variations into a karyotype graph representation of the rearranged cancer genomes. Such a representation, however, does not directly describe the linear and/or circular structure of the underlying rearranged cancer chromosomes, thus limiting possible analysis of cancer genomes somatic evolutionary process as well as functional genomic changes brought by the large-scale genome rearrangements.ResultsHere we address the aforementioned limitation by introducing a novel methodological framework for recovering rearranged cancer chromosomes from karyotype graphs. For a cancer karyotype graph we formulate an Eulerian Decomposition Problem (EDP) of finding a collection of linear and/or circular rearranged cancer chromosomes that are determined by the graph. We derive and prove computational complexities for several variations of the EDP. We then demonstrate that Eulerian decomposition of the cancer karyotype graphs is not always unique and present the Consistent Contig Covering Problem (CCCP) of recovering unambiguous cancer contigs from the cancer karyotype graph, and describe a novel algorithm CCR capable of solving CCCP in polynomial time. We apply CCR on a prostate cancer dataset and demonstrate that it is capable of consistently recovering large cancer contigs even when underlying cancer genomes are highly rearranged.ConclusionsCCR can recover rearranged cancer contigs from karyotype graphs thereby addressing existing limitation in inferring chromosomal structures of rearranged cancer genomes and advancing our understanding of both patient/cancer-specific as well as the overall genetic instability in cancer.

On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2 -factor F 1 and s copies of a 2 -factor F 2 such that r + s = ⌊( v − 1) / 2⌋ . If F 1 consists of m -cycles and F 2 consists of n cycles, we say that a solution to ( m , n ) - HWP( v ; r , s ) exists. The goal is to find a decomposition for every possible pair ( r , s ) . In this paper, we show that for odd x and y , there is a solution to (2 k x , y ) - HWP( v m ; r , s ) if gcd ( x , y ) ≥ 3 , m ≥ 3 , and both x and y divide v , except possibly when 1 ∈ { r , s } .

Decomposition of complete graphs into connected unicyclic bipartite graphs with eight edges

We prove that each of the 34 non-isomorphic connected unicyclic bipartite graphs with eight edges decomposes the complete graph K n whenever the necesary conditions are satisfied.

A Faster Tree-Decomposition Based Algorithm for Counting Linear Extensions

We investigate the problem of computing the number of linear extensions of a given n-element poset whose cover graph has treewidth t. We present an algorithm that runs in time {tilde{O}}(n^{t+3}) for any constant t; the notation {tilde{O}} hides polylogarithmic factors. Our algorithm applies dynamic programming along a tree decomposition of the cover graph; the join nodes of the tree decomposition are handled by fast multiplication of multivariate polynomials. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone: fixing these parameters leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. We compare two approaches to select an efficient tree decomposition: one is to include additional features of the tree decomposition to build a more accurate, heuristic cost function; the other approach is to fit a statistical regression model to collected running time data. Both approaches are shown to yield a tree decomposition that typically is significantly more efficient than a random optimal-width tree decomposition.

Faster algorithms for shortest path and network flow based on graph decomposition

We propose faster algorithms for the maximum flow problem and shortest path problems based on graph decomposition. Our algorithms first construct indices (data structures) from a given graph, then use them for solving the problems. Time complexities of our algorithms depend on the size of the maximum triconnected component in the graph, say $r$. Max flow indexing problem is a basic network flow problem, which consists of two phases. In a preprocessing phase we construct an index and in a query phase we process the query using the index. We can solve all pairs maximum flow problem and minimum cut problem using the indices. Our algorithms run faster than known algorithms if $r$ is small. The maximum flow problem can be solved in $\mathcal{O}(nr)$ time, which is faster than the best known $\mathcal{O}(nm)$ algorithm [Orlin, STOC 2013] if $r = o(m)$, where $n$ and $m$ are the numbers of vertices and edges in the given network, respectively. Distance oracle problem is a basic problem in shortest path, consisting of two phases. In preprocessing phase we construct index and in query phase we use the index to find shortest path between two vertices. We use these indices to solve single source shortest path and all pair shortest path problems. If the given graph is undirected and all the weights are non-negative integers, then our algorithm finds shortest path between two vertices in $\mathcal{O}(m)$ time. If the given graph is directed or the weights are non-negative real numbers then our algorithm finds shortest path between two vertices in $\mathcal{O}(m + n \log r)$ time. If the edge weights are real numbers (i.e some of the weights are negative) then our algorithm finds shortest path between two vertices in $\mathcal{O}(m + nr)$ time.

On the minimum number of bond-edge types and tile types: An approach by edge-colorings of graphs

The purpose of this paper is twofold. On one hand, we want to describe from a new graph theory perspective the self-assembly of DNA structures with branched junction molecules having flexible arms. On the other hand, we employ edge-colorings and graph decompositions to study the well-known problem of determining the minimum number of bond-edge types and tile types, which are graph invariants appearing in this context. We provide a strategy that can be applied to arbitrary graphs for obtaining upper bounds for these graph invariants.

Density-Friendly Graph Decomposition

Decomposing a graph into a hierarchical structure via k -core analysis is a standard operation in any modern graph-mining toolkit. k -core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph. Despite the fact that k -core analysis relies on vertex degrees, k -cores do not satisfy a certain, rather natural, density property. Simply put, the most central k -core is not necessarily the densest subgraph. This inconsistency between k -cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally dense , and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by k -cores, but in this case the components are arranged in order of increasing density. We show that such a locally dense decomposition for a graph G =( V , E ) can be computed in polynomial time. The running time of the exact decomposition algorithm is O (| V | 2 | E |) but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally dense decomposition. Furthermore, we show that the k -core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice k -cores have different structure than locally dense subgraphs, and as predicted by the theory, k -cores are not always well-aligned with graph density.

Shorter tours and longer detours: uniform covers and a bit beyond

Motivated by the well known “four-thirds conjecture” for the traveling salesman problem (TSP), we study the problem of uniform covers. A graph $$G=(V,E)$$ has an $$\alpha $$ -uniform cover for TSP (2EC, respectively) if the everywhere $$\alpha $$ vector (i.e., $$\{\alpha \}^{E}$$ ) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides’ algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Sebő asked if such graphs have $$(1-\epsilon )$$ -uniform covers for TSP for some $$\epsilon > 0$$ . Indeed, the four-thirds conjecture implies that such graphs have $$\frac{8}{9}$$ -uniform covers. We show that these graphs have $$\frac{18}{19}$$ -uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere $$\frac{15}{17}$$ vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere $$\frac{2}{3}$$ vector is an optimal solution for the subtour elimination linear programming relaxation for TSP, then a tour with weight at most $$\frac{27}{19}$$ times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. An essential ingredient in our proofs is decompositions of graphs (e.g., cycle covers) that cover small-cardinality cuts an even (nonzero) number of times. Another essential tool we use is half-integral tree augmentation, which is known to have a small integrality gap. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose a point in the subtour elimination polytope into spanning, connected subgraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a $$\frac{17}{12}$$ -approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs.

Quadratic Vertex Kernel for Rainbow Matching

In this paper, we study the NP-complete colorful variant of the classic matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, the goal is to decide whether there exists a matching of size at least k such that the edges in the matching have distinct colors. Previously, in [MFCS’17], we studied this problem from the view point of Parameterized Complexity and gave efficient FPT algorithms as well as a quadratic kernel on paths. In this paper we design a quadratic vertex kernel for Rainbow Matching on general graphs; generalizing the earlier quadratic kernel on paths to general graphs. For our kernelization algorithm we combine a graph decomposition method with an application of expansion lemma.

Optimal packings of bounded degree trees

We prove that if T_1,\dots, T_n is a sequence of bounded degree trees such that T_i has i vertices, then K_n has a decomposition into T_1,\dots, T_n . This shows that the tree packing conjecture of Gyárfás and Lehel from 1976 holds for all bounded degree trees (in fact, we can allow the first o(n) trees to have arbitrary degrees). Similarly, we show that Ringel's conjecture from 1963 holds for all bounded degree trees. We deduce these results from a more general theorem, which yields decompositions of dense quasi-random graphs into suitable families of bounded degree graphs. Our proofs involve Szemerédi's regularity lemma, results on Hamilton decompositions of robust expanders, random walks, iterative absorption as well as a recent blow-up lemma for approximate decompositions.

Decompositions of some regular graphs into unicyclic graphs of order five

For a graph [Formula: see text] and a subgraph [Formula: see text] of [Formula: see text] an [Formula: see text]-decomposition of [Formula: see text] is a partition of the edge set of [Formula: see text] into subsets [Formula: see text] [Formula: see text] such that each [Formula: see text] induces a graph isomorphic to [Formula: see text] It is proved that the necessary conditions are sufficient for the existence of an [Formula: see text]-decomposition of the graph [Formula: see text] where [Formula: see text] is any simple connected unicyclic graph of order five, × denotes the tensor product of graphs and [Formula: see text] denotes the multiplicity of the edges. In fact, using the above characterization, a necessary and sufficient condition for the graph [Formula: see text] [Formula: see text] and [Formula: see text] to admit an [Formula: see text]-decomposition is obtained. Similar results for the complete graphs and complete multipartite graphs are proved in: [J.-C. Bermond et al. [Formula: see text]-decomposition of [Formula: see text], where [Formula: see text] has four vertices or less, Discrete Math. 19 (1977) 113–120, J.-C. Bermond et al. Decomposition of complete graphs into isomorphic subgraphs with five verices, Ars Combin. 10 (1980) 211–254, M. H. Huang, Decomposing complete equipartite graphs into connected unicyclic graphs of size five, Util. Math. 97 (2015) 109–117].

Grammars and clique-width bounds from split decompositions

Graph decompositions are important for algorithmic purposes and for graph structure theory. We relate the split decomposition introduced by Cunnigham to vertex substitution, graph grammars and clique-width.For this purpose, we extend the usual notion of substitution, upon which modular decomposition is based, by considering graphs with dead (or non-boundary) vertices. We obtain a graph grammar for distance-hereditary graphs consisting of four rules collected in a single equation. We also bound the clique-width of a graph in terms of those of the components of a split decomposition that need not be canonical.For extending these results to directed graphs and their split decompositions (that we handle formally as graph-labelled trees), we need another extension of substitution : instead of two types of vertices, dead or alive as for undirected graphs, we need four types, in order to encode edge directions. We bound linearly the clique-width of a directed graph G in terms of the maximal clique-width of a component arising in a graph-labelled tree that defines G. This result concerns all directed graphs, not only the strongly connected ones considered by Cunningham.

Continuous Monotonic Decomposition of Some Standard Graphs by using an Algorithm

In this paper we elaborate an algorithm to compute the necessary and sufficient conditions for the continuous monotonic star decomposition of the bipartite graph Km,r and the number of vertices and the number of disjoint sets. Also an algorithm to find the tensor product of Pn  Ps has continuous monotonic path decomposition. Finally we conclude that in this paper the results described above are complete bipartite graphs that accept Continuous monotonic star decomposition. There are many other classes of complete tripartite graphs that accept Continuous monotonic star decomposition. In this research article Extended to complete m-partite graphs for grater values of m. Also the algorithm can be developed for the tensor product of different classes such as Cn Wn K1,n , , with Pn

Fast Graph Fourier Transforms Based on Graph Symmetry and Bipartition

The graph Fourier transform (GFT) is an important tool for graph signal processing, with applications ranging from graph-based image processing to spectral clustering. However, unlike the discrete Fourier transform, the GFT typically does not have a fast algorithm. In this work, we develop new approaches to accelerate the GFT computation. In particular, we show that Haar units (Givens rotations with angle π/4) can be used to reduce GFT computation cost when the graph is bipartite or satisfies certain symmetry properties based on node pairing. We also propose a graph decomposition method based on graph topological symmetry, which allows us to identify and exploit butterfly structures in stages. This method is particularly useful for graphs that are nearly regular or have some specific structures, e.g., line graphs, cycle graphs, grid graphs, and human skeletal graphs. Though butterfly stages based on graph topological symmetry cannot be used for general graphs, they are useful in applications, including video compression and human action analysis, where symmetric graphs, such as symmetric line graphs and human skeletal graphs, are used. Our proposed fast GFT implementations are shown to reduce computation costs significantly, in terms of both number of operations and empirical runtimes.

Plan-Length Bounds: Beyond 1-Way Dependency

We consider the problem of compositionally computing upper bounds on lengths of plans. Following existing work, our approach is based on a decomposition of state-variable dependency graphs (a.k.a. causal graphs). Tight bounds have been demonstrated previously for problems where key dependencies flow in a single direction—i.e. manipulating variable v1 can disturb the ability to manipulate v2 and not vice versa. We develop a more general bounding approach which allows us to compute useful bounds where dependency flows in both directions. Our approach is practically most useful when combined with earlier approaches, where the computed bounds are substantially improved in a relatively broad variety of problems. When combined with an existing planning procedure, the improved bounds yield coverage improvements for both solvable and unsolvable planning problems.

Constrained ear decompositions in graphs and digraphs

Ear decompositions of graphs are a standard concept related to several major problems in graph theory like the Traveling Salesman Problem. For example, the Hamiltonian Cycle Problem, which is notoriously N P-complete, is equivalent to deciding whether a given graph admits an ear decomposition in which all ears except one are trivial (i.e. of length 1). On the other hand, a famous result of Lovasz states that deciding whether a graph admits an ear decomposition with all ears of odd length can be done in polynomial time. In this paper, we study the complexity of deciding whether a graph admits an ear decomposition with prescribed ear lengths. We prove that deciding whether a graph admits an ear decomposition with all ears of length at most is polynomial-time solvable for all fixed positive integer. On the other hand, deciding whether a graph admits an ear decomposition without ears of length in F is N P-complete for any finite set F of positive integers. We also prove that, for any k ≥ 2, deciding whether a graph admits an ear decomposition with all ears of length 0 mod k is N P-complete. We also consider the directed analogue to ear decomposition, which we call handle decomposition, and prove analogous results : deciding whether a digraph admits a handle decomposition with all handles of length at most is polynomial-time solvable for all positive integer ; deciding whether a digraph admits a handle decomposition without handles of length in F is N P-complete for any finite set F of positive integers (and minimizing the number of handles of length in F is not approximable up to n(1 −)); for any k ≥ 2, deciding whether a digraph admits a handle decomposition with all handles of length 0 mod k is N P-complete. Also, in contrast with the result of Lovasz, we prove that deciding whether a digraph admits a handle decomposition with all handles of odd length is N P-complete. Finally, we conjecture that, for every set A of integers, deciding whether a digraph has a handle decomposition with all handles of length in A is N P-complete, unless there exists h ∈ N such that A = {1, · · · , h}.

Embedding an edge‐coloring of K(nr;λ1,λ2) into a Hamiltonian decomposition of K(nr+2;λ1,λ2)

AbstractThis paper focuses on graph decompositions of , the ‐partite multigraph in which each part has size , where two vertices in the same part or different parts are joined by exactly edges or edges respectively. Assuming one condition, necessary and sufficient conditions are found to embed a k‐edge‐coloring of into a Hamiltonian decomposition of . In the tightest case, this assumption is in fact proved to be a new necessary condition. Unlike previous results, of particular interest here is a necessary condition involving the existence of certain components in a related bipartite graph.