Faster Sublinear Approximation of the Number of K-Cliques in Low-Arboricity Graphs
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Linear K-Arboricities on Trees
Discrete Applied Mathematics 103 (2000) 281–287 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Note Linear k-arboricities on trees Gerard J. Changa; 1, Bor-Liang Chenb, Hung-Lin Fua, Kuo-Ching Huangc; ∗;2 aDepartment of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan bDepartment of Business Administration, National Taichung Institue of Commerce, Taichung 404, Taiwan cDepartment of Applied Mathematics, Providence University, Shalu 433, Taichung, Taiwan Received 31 October 1997; revised 15 November 1999; accepted 22 November 1999 Abstract For a ÿxed positive integer k, the linear k-arboricity lak (G) of a graph G is the minimum number ‘ such that the edge set E(G) can be partitioned into ‘ disjoint sets and that each induces a subgraph whose components are paths of lengths at most k. This paper studies linear k-arboricity from an algorithmic point of view. In particular, we present a linear-time algo- rithm to determine whether a tree T has lak (T)6m. ? 2000 Elsevier Science B.V. All rights reserved. Keywords: Linear forest; Linear k-forest; Linear arboricity; Linear k-arboricity; Tree; Leaf; Penultimate vertex; Algorithm; NP-complete 1. Introduction All graphs in this paper are simple, i.e., ÿnite, undirected, loopless, and without multiple edges. A linear k-forest is a graph whose components are paths of length at most k.Alinear k-forest partition of G is a partition of the edge set E(G) into linear k-forests. The linear k-arboricity of G, denoted by lak (G), is the minimum size of a linear k-forest partition of G. -
Orienting Fully Dynamic Graphs with Worst-Case Time Bounds⋆
Orienting Fully Dynamic Graphs with Worst-Case Time Bounds? Tsvi Kopelowitz1??, Robert Krauthgamer2 ???, Ely Porat3, and Shay Solomon4y 1 University of Michigan, [email protected] 2 Weizmann Institute of Science, [email protected] 3 Bar-Ilan University, [email protected] 4 Weizmann Institute of Science, [email protected] Abstract. In edge orientations, the goal is usually to orient (direct) the edges of an undirected network (modeled by a graph) such that all out- degrees are bounded. When the network is fully dynamic, i.e., admits edge insertions and deletions, we wish to maintain such an orientation while keeping a tab on the update time. Low out-degree orientations turned out to be a surprisingly useful tool for managing networks. Brodal and Fagerberg (1999) initiated the study of the edge orientation problem in terms of the graph's arboricity, which is very natural in this context. Their solution achieves a constant out-degree and a logarithmic amortized update time for all graphs with constant arboricity, which include all planar and excluded-minor graphs. It remained an open ques- tion { first proposed by Brodal and Fagerberg, later by Erickson and others { to obtain similar bounds with worst-case update time. We address this 15 year old question by providing a simple algorithm with worst-case bounds that nearly match the previous amortized bounds. Our algorithm is based on a new approach of maintaining a combina- torial invariant, and achieves a logarithmic out-degree with logarithmic worst-case update times. This result has applications to various dynamic network problems such as maintaining a maximal matching, where we obtain logarithmic worst-case update time compared to a similar amor- tized update time of Neiman and Solomon (2013). -
On Star and Caterpillar Arboricity
Discrete Mathematics 309 (2009) 3694–3702 www.elsevier.com/locate/disc On star and caterpillar arboricity Daniel Gonc¸alvesa,∗, Pascal Ochemb a LIRMM UMR 5506, CNRS, Universite´ Montpelier 2, 161 rue Ada, 34392 Montpellier Cedex 5, France b LRI UMR 8623, CNRS, Universite´ Paris-Sud, Batˆ 490, 91405 Orsay Cedex, France Received 31 October 2005; accepted 18 January 2008 Available online 10 March 2008 Abstract We give new bounds on the star arboricity and the caterpillar arboricity of planar graphs with given girth. One of them answers an open problem of Gyarf´ as´ and West: there exist planar graphs with track number 4. We also provide new NP-complete problems. c 2008 Elsevier B.V. All rights reserved. Keywords: NP-completeness; Partitioning problems; Edge coloring 1. Introduction Many graph parameters in the literature are defined as the minimum size of a partition of the edges of the graph such that each part induces a graph of a given class C. The most common is the chromatic index χ 0.G/, in this case C is the class of graphs with maximum degree one. Vizing [18] proved that χ 0.G/ either equals ∆.G/ or ∆.G/ C 1, where ∆.G/ denotes the maximum degree of G. Deciding whether χ 0.G/ D 3 is shown to be NP-complete for general graphs in [13]. The arboricity a.G/ is another well studied parameter, for which C is the class of forests. In [15], Nash-Williams proved that: jE.H/j a.G/ D max (1) H⊆G jV .H/j − 1 with the maximum being over all the subgraphs H D .E.H/; V .H// of G. -
Graph Treewidth and Geometric Thickness Parameters
Graph Treewidth and Geometric Thickness Parameters Vida Dujmovi´c 1? and David R. Wood 2?? 1 School of Computer Science, Carleton University, Ottawa, Canada ([email protected]) 2 Departament de Matem`atica Aplicada II, Universitat Polit`ecnicade Catalunya, Barcelona, Spain ([email protected]) Abstract. Consider a drawing of a graph G in the plane such that cross- ing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness θ(G). By restricting the edges to be straight, we obtain the geometric thick- ness θ(G). By further restricting the vertices to be in convex position, we obtain the book thickness bt(G). This paper studies the relationship between these parameters and the treewidth of G. Let θ(Tk)/ θ(Tk) / bt(Tk) denote the maximum thickness / geometric thickness / book thickness of a graph with treewidth at most k. We prove that: – θ(Tk) = θ(Tk) = dk/2e, and – bt(Tk) = k for k ≤ 2, and bt(Tk) = k + 1 for k ≥ 3. The first result says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. The second result disproves the conjecture of Ganley and Heath [Discrete Appl. Math. 2001] that bt(Tk) = k for all k. Analogous results are proved for outerthickness, arboricity, and star-arboricity. 1 Introduction Partitions of the edge set of a graph G into a small number of ‘nice’ subgraphs is in the mainstream of graph theory. For example, in a proper edge colouring, the subgraphs of the partition are matchings. -
Decomposition of Sparse Graphs Into Forests and a Graph with Bounded Degree
Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree Seog-Jin Kim∗, Alexandr V. Kostochka† , Douglas B. West‡, Hehui Wu§, and Xuding Zhu¶ Revised October, 2012 Abstract For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum |E(H)| of |V (H)|−1 over all subgraphs H with at least two vertices. Generalizing the Nash- Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if Arb(G) ≤ d k + k+d+1 , then G decomposes into k + 1 forests with one having maximum degree at most d. The conjecture was previously proved for (k, d) ∈ {(1, 1), (1, 2)}; we prove it for d = k +1 and when k = 1 and d ≤ 6. For (k, d) = (1, 2), we can further restrict one forest to have at most two edges in each component. d For general (k, d), we prove weaker conclusions. If d > k, then Arb(G) ≤ k + k+d+1 implies that G decomposes into k forests plus a multigraph (not necessarily a forest) d with maximum degree at most d. If d ≤ k, then Arb(G) ≤ k + 2k+2 implies that G decomposes into k + 1 forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs. 1 Introduction Throughout this paper, we use the model of “graph” in which multiedges but no loops are allowed. A decomposition of a graph G consists of edge-disjoint subgraphs with union G. The arboricity of G, written Υ(G), is the minimum number of forests needed to decompose it. -
On the Vertex-Arboricity of Planar Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector European Journal of Combinatorics 29 (2008) 1064–1075 www.elsevier.com/locate/ejc On the vertex-arboricity of planar graphs Andre´ Raspaud, Weifan Wang1 LaBRI UMR CNRS 5800, Universite´ Bordeaux I, 33405 Talence Cedex, France Available online 24 January 2008 Abstract The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G) ≤ 3 for any planar graph G. In this paper we prove that a(G) ≤ 2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3. c 2007 Elsevier Ltd. All rights reserved. 1. Introduction All graphs considered in this paper are finite simple graphs. A plane graph is a particular drawing of a planar graph on the Euclidean plane. For a plane graph G, let V (G), E(G), F(G), |G|, G∗, ∆(G), and δ(G) denote, respectively, its vertex set, edge set, face set, order, dual, maximum degree, and minimum degree. A linear forest is a forest in which every connected component is a path. The vertex-arboricity a(G) (linear vertex-arboricity la(G), respectively) of a graph G is the minimum number of subsets into which V (G) can be partitioned so that each subset induces a forest (a linear forest, respectively). -
STRUCTURAL RESULTS on MAXIMAL K-DEGENERATE GRAPHS
Discussiones Mathematicae Graph Theory 32 (2012) 659–676 doi:10.7151/dmgt.1637 STRUCTURAL RESULTS ON MAXIMAL k-DEGENERATE GRAPHS Allan Bickle Department of Mathematics Western Michigan University 1903 W. Michigan Kalamazoo, MI 49008 e-mail: [email protected] Abstract A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. These graphs were introduced by Lick and White in 1970 and have been studied in several subsequent papers. We present sharp bounds on the diameter of maximal k-degenerate graphs and characterize the extremal graphs for the upper bound. We present a simple characterization of the degree sequences of these graphs and consider related results. Considering edge coloring, we conjecture that a maximal k-degenerate graph is class two if and only if it is overfull, and prove this in some special cases. We present some results on decompositions and arboric- ity of maximal k-degenerate graphs and provide two characterizations of the subclass of k-trees as maximal k-degenerate graphs. Finally, we define and prove a formula for the Ramsey core numbers. Keywords: k-degenerate, k-core, k-tree, degree sequence, Ramsey number. 2010 Mathematics Subject Classification: 05C75. 1. Introduction One of the basic properties of graphs is the existence of subgraphs with specified degree conditions. (See [3] and [18] for basic terminology.) Definition. The k-core of a graph G, Ck (G), is the maximal induced subgraph H G such that δ(H) k, if it exists. ⊆ ≥ Cores were introduced by S.B. -
On Graph Thickness, Geometric Thickness, and Separator Theorems
CCCG 2009, Vancouver, BC, August 17{19, 2009 On Graph Thickness, Geometric Thickness, and Separator Theorems Christian A. Duncan∗ Abstract fined to be the smallest number of forests whose union is G [25, 26, 27]. In other words, the minimum number We investigate the relationship between geometric of edge colors such that the subgraph induced by each thickness and the thickness, outerthickness, and ar- color is a forest of trees. In linear arboricity the result- boricity of graphs. In particular, we prove that all ing colored subgraphs must be collections of paths [2, 3]. graphs with arboricity two or outerthickness two have In outerthickness, the resulting colored subgraphs must geometric thickness O(log n). The technique used can be outerplanar graphs [16, 17]. Outerplanar graphs are be extended to other classes of graphs so long as a stan- analogous to graphs with book thickness one; however, dard separator theorem exists. For example, we can outerthickness and book thickness are not identical. apply it to show the known boundp that thickness two Another way to look at the problem is to divide the graphs have geometric thickness O( n), yielding a sim- edges of the graphs into different layers and draw the ple construction in the process. layers independently as planar graphs such that the ver- tex positions are identical in each layer. In this case, 1 Introduction the problem is to minimize the number of layers. A common related application of this problem is in VLSI In many applications of graph visualization and graph design where wires are placed in various layers to avoid theory, it is often useful to draw the edges of a graph crossings, see for example [1]. -
A Tutorial on Clique Problems in Communications and Signal Processing Ahmed Douik, Student Member, IEEE, Hayssam Dahrouj, Senior Member, IEEE, Tareq Y
1 A Tutorial on Clique Problems in Communications and Signal Processing Ahmed Douik, Student Member, IEEE, Hayssam Dahrouj, Senior Member, IEEE, Tareq Y. Al-Naffouri, Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE Abstract—Since its first use by Euler on the problem of the be found in the following reference [4]. The textbooks by seven bridges of Konigsberg,¨ graph theory has shown excellent Bollobas´ [5] and Diestel [6] provide an in-depth investigation abilities in solving and unveiling the properties of multiple of modern graph theory tools including flows and connectivity, discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory the coloring problem, random graphs, and trees. problems making a large body of the literature readily available Current applications of graph theory span several fields. for solving and characterizing the complexity of these problems. Indeed, graph theory, as a part of discrete mathematics, is This tutorial presents a framework for utilizing a particular particularly helpful in solving discrete equations with a well- graph theory problem, known as the clique problem, for solving defined structure. Reference [7] provides multiple connections communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer between graph theory problems and their combinatoric coun- programs that can be formulated as clique problems through terparts. Multiple tutorials on the applications of graph theory multiple examples in communications and signal processing. To techniques to various problems are available in the literature, that end, the first part of the tutorial provides various optimal e.g., [8]–[15]. -
Induced and Weak Induced Arboricities
Induced and Weak Induced Arboricities Maria Axenovich, Philip D¨orr,Jonathan Rollin, Torsten Ueckerdt February 26, 2018 Abstract We define the induced arboricity of a graph G, denoted by ia(G), as the smallest k such that the edges of G can be covered with k induced forests in G. This notion generalizes the classical notions of the arboricity and strong chromatic index. For a class F of graphs and a graph parameter p, let p(F) = supfp(G) j G 2 Fg. We show that ia(F) is bounded from above by an abso- lute constant depending only on F, that is ia(F) 6= 1 if and only if χ(Fr1=2) 6= 1, where Fr1=2 is the class of 1=2-shallow minors of graphs from F and χ is the chromatic number. Further, we give bounds on ia(F) when F is the class of planar graphs, the class of d-degenerate graphs, or the class of graphs having tree-width at most d. Specifically, we show that if F is the class of planar graphs, then 8 ≤ ia(F) ≤ 10. In addition, we establish similar results for so-called weak induced arboricities and star arboricities of classes of graphs. 1 Introduction For a graph G, the arboricity a(G) is the smallest number of forests covering all the edges of G. As Nash-Williams proved 50 years ago, the arboricity is governed precisely by the largest density among the subgraphs of G. Let jE(H)j m(G) = max : H ⊆ G; jV (H)j ≥ 2 : jV (H)j − 1 Theorem 1 (Nash-Williams [12]). -
Arxiv Papers on Random Structures
Arxiv Papers on Random Structures April 24, 2020 1 Papers 1. Random Algebra (a) Random Relation Algebras 2. Random Complexes 1 Random Uniform and Pure Random Simplicial Complexes 2 A limit theorem for the 1st Betti number of layer-1 subgraphs in random graphs 3 Homotopy Types of Random Cubical Complexes 4 Dynamical Models for Random Simplicial Complexes 5 A Sharp Threshold for Spanning 2-Spheres in Random 2-Complexes 6 Topology of random geometric complexes: a survey 7 A Model for Random Chain Complexes 8 Random Chain Complexes 9 Topology of random clique complexes 10 Topology of random d-clique complexes 11 Vanishing of cohomology groups of random simplicial complexes 12 Collapsibility of Random Clique Complexes 13 Random Simplicial Complexes, Duality and The Critical Dimension 14 On the Cycle Space of a Random Graph 15 Thresholds for vanishing of `Isolated' faces in random Cech and Vietoris-Rips complexes 16 On freeness of the random fundamental group 17 Gigantic random simplicial complexes 18 A Sharp Threshold for Spanning 2-Spheres in Random 2-Complexes 19 Higher dimensional connectivity and minimal degree of random graphs with an eye towards minimal free resolutions 20 Homological connectivity of random hypergraphs 21 On the Vanishing of Homology in Random Cech Complexes 1 22 Integral homology of random simplicial complexes 23 The integer homology threshold in Yd(n; p) 24 On persistent homology of random Cech complexes 25 On the phase transition in random simplicial complexes 26 On simple connectivity of random 2-complexes 27 Random Steiner systems and bounded degree coboundary expanders of every dimension 28 Topology of random simplicial complexes: a survey 29 Spectral bounds for vanishing of cohomology and the neighborhood complex of a random graph 30 Randomly Weighted dcomplexes: Minimal Spanning Acycles and Persistence Diagrams 31 Random Simplicial Complexes in the Medial Regime 32 Asymptotic behavior of lifetime sums for random simplicial complex processes 3. -
K-Degenerate Graphs
k-Degenerate Graphs Allan Bickle Western Michigan University Date Allan Bickle The k-Cores of a Graph Basics Denition The k-core of a graph G is the maximal induced subgraph H ⊆ G such that d (H) ≥ k. The core number of a vertex, C (v), is the largest value for k such that v 2 Ck (G). The maximum core number of a graph, Cb (G), is the maximum of the core numbers of the vertices of G. If the maximum core number and minimum degree of G are equal, Cb (G) = d (G), we say G is k-monocore. Allan Bickle The k-Cores of a Graph Basics The k-core algorithm (sketch) Input: graph G with adjacency matrix A, integer k, degree array D. Recursion: Delete all vertices with degree less than k from G. (That is, make a list of such vertices, zero out their degrees, and decrement the degrees of their neighbors.) Result: The vertices that have not been deleted induce the k-core. Allan Bickle The k-Cores of a Graph Basics Theorem Applying the k-core algorithm to graph G yields the k-core of G, provided it exists. Denition A vertex deletion sequence of a graph G is a sequence that contains each of its vertices exactly once and is formed by successively deleting a vertex of smallest degree. Allan Bickle The k-Cores of a Graph Basics Denition A graph is k-degenerate if its vertices can be successively deleted so that when deleted, each has degree at most k. The degeneracy of a graph is the smallest k such that it is k-degenerate.