An Updated Survey on Rainbow Connections of Graphs - a Dynamic Survey Xueliang Li Nankai University, China, [email protected]
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On the Tree–Depth of Random Graphs Arxiv:1104.2132V2 [Math.CO] 15 Feb 2012
On the tree–depth of random graphs ∗ G. Perarnau and O.Serra November 11, 2018 Abstract The tree–depth is a parameter introduced under several names as a measure of sparsity of a graph. We compute asymptotic values of the tree–depth of random graphs. For dense graphs, p n−1, the tree–depth of a random graph G is a.a.s. td(G) = n − O(pn=p). Random graphs with p = c=n, have a.a.s. linear tree–depth when c > 1, the tree–depth is Θ(log n) when c = 1 and Θ(log log n) for c < 1. The result for c > 1 is derived from the computation of tree–width and provides a more direct proof of a conjecture by Gao on the linearity of tree–width recently proved by Lee, Lee and Oum [?]. We also show that, for c = 1, every width parameter is a.a.s. constant, and that random regular graphs have linear tree–depth. 1 Introduction An elimination tree of a graph G is a rooted tree on the set of vertices such that there are no edges in G between vertices in different branches of the tree. The natural elimination scheme provided by this tree is used in many graph algorithmic problems where two non adjacent subsets of vertices can be managed independently. One good example is the Cholesky decomposition of symmetric matrices (see [?, ?, ?, ?]). Given an elimination tree, a distributed algorithm can be designed which takes care of disjoint subsets of vertices in different parallel processors. Starting arXiv:1104.2132v2 [math.CO] 15 Feb 2012 by the furthest level from the root, it proceeds by exposing at each step the vertices at a given depth. -
On the Density of a Graph and Its Blowup
On the Density of a Graph and its Blowup Asaf Shapira ¤ Raphael Yustery Abstract It is well-known that, of all graphs with edge-density p, the random graph G(n; p) contains the smallest density of copies of Kt;t, the complete bipartite graph of size 2t. Since Kt;t is a t-blowup of an edge, the following intriguing open question arises: Is it true that of all graphs with triangle 3 density p , the random graph G(n; p) contains close to the smallest density of Kt;t;t, which is the t-blowup of a triangle? Our main result gives an indication that the answer to the above question is positive by showing that for some blowup, the answer must be positive. More formally we prove that if G 3 has triangle density p , then there is some 2 · t · T (p) for which the density of Kt;t;t in G is at (3+o(1))t2 least p , which (up to the o(1) term) equals the density of Kt;t;t in G(n; p). We also raise several open problems related to these problems and discuss some applications to other areas. 1 Introduction One of the main family of problems studied in extremal graph theory is how does the lack and/or number of copies of one graph H in a graph G a®ect the lack and/or number of copies of another graph H0 in G. Perhaps the most well known problems of this type are Ramsey's Theorem and Tur¶an'sTheorem. -
Testing Isomorphism in the Bounded-Degree Graph Model (Preliminary Version)
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 102 (2019) Testing Isomorphism in the Bounded-Degree Graph Model (preliminary version) Oded Goldreich∗ August 11, 2019 Abstract We consider two versions of the problem of testing graph isomorphism in the bounded-degree graph model: A version in which one graph is fixed, and a version in which the input consists of two graphs. We essentially determine the query complexity of these testing problems in the special case of n-vertex graphs with connected components of size at most poly(log n). This is done by showing that these problems are computationally equivalent (up to polylogarithmic factors) to corresponding problems regarding isomorphism between sequences (over a large alphabet). Ignoring the dependence on the proximity parameter, our main results are: 1. The query complexity of testing isomorphism to a fixed object (i.e., an n-vertex graph or an n-long sequence) is Θ(e n1=2). 2. The query complexity of testing isomorphism between two input objects is Θ(e n2=3). Testing isomorphism between two sequences is shown to be related to testing that two distributions are equivalent, and this relation yields reductions in three of the four relevant cases. Failing to reduce the problem of testing the equivalence of two distribution to the problem of testing isomorphism between two sequences, we adapt the proof of the lower bound on the complexity of the first problem to the second problem. This adaptation constitutes the main technical contribution of the current work. Determining the complexity of testing graph isomorphism (in the bounded-degree graph model), in the general case (i.e., for arbitrary bounded-degree graphs), is left open. -
Practical Verification of MSO Properties of Graphs of Bounded Clique-Width
Practical verification of MSO properties of graphs of bounded clique-width Ir`ene Durand (joint work with Bruno Courcelle) LaBRI, Universit´ede Bordeaux 20 Octobre 2010 D´ecompositions de graphes, th´eorie, algorithmes et logiques, 2010 Objectives Verify properties of graphs of bounded clique-width Properties ◮ connectedness, ◮ k-colorability, ◮ existence of cycles ◮ existence of paths ◮ bounds (cardinality, degree, . ) ◮ . How : using term automata Note that we consider finite graphs only 2/33 Graphs as relational structures For simplicity, we consider simple,loop-free undirected graphs Extensions are easy Every graph G can be identified with the relational structure (VG , edgG ) where VG is the set of vertices and edgG ⊆VG ×VG the binary symmetric relation that defines edges. v7 v6 v2 v8 v1 v3 v5 v4 VG = {v1, v2, v3, v4, v5, v6, v7, v8} edgG = {(v1, v2), (v1, v4), (v1, v5), (v1, v7), (v2, v3), (v2, v6), (v3, v4), (v4, v5), (v5, v8), (v6, v7), (v7, v8)} 3/33 Expression of graph properties First order logic (FO) : ◮ quantification on single vertices x, y . only ◮ too weak ; can only express ”local” properties ◮ k-colorability (k > 1) cannot be expressed Second order logic (SO) ◮ quantifications on relations of arbitrary arity ◮ SO can express most properties of interest in Graph Theory ◮ too complex (many problems are undecidable or do not have a polynomial solution). 4/33 Monadic second order logic (MSO) ◮ SO formulas that only use quantifications on unary relations (i.e., on sets). ◮ can express many useful graph properties like connectedness, k-colorability, planarity... Example : k-colorability Stable(X ) : ∀u, v(u ∈ X ∧ v ∈ X ⇒¬edg(u, v)) Partition(X1,..., Xm) : ∀x(x ∈ X1 ∨ . -
Harary Polynomials
numerative ombinatorics A pplications Enumerative Combinatorics and Applications ECA 1:2 (2021) Article #S2R13 ecajournal.haifa.ac.il Harary Polynomials Orli Herscovici∗, Johann A. Makowskyz and Vsevolod Rakitay ∗Department of Mathematics, University of Haifa, Israel Email: [email protected] zDepartment of Computer Science, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] yDepartment of Mathematics, Technion{Israel Institute of Technology, Haifa, Israel Email: [email protected] Received: November 13, 2020, Accepted: February 7, 2021, Published: February 19, 2021 The authors: Released under the CC BY-ND license (International 4.0) Abstract: Given a graph property P, F. Harary introduced in 1985 P-colorings, graph colorings where each color class induces a graph in P. Let χP (G; k) counts the number of P-colorings of G with at most k colors. It turns out that χP (G; k) is a polynomial in Z[k] for each graph G. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial χ(G; k). We show that the characteristic and the Laplacian polynomial, the matching, the independence and the domination polynomials are not Harary polynomials. We show that for various notions of sparse, non-trivial properties P, the polynomial χP (G; k) is, in contrast to χ(G; k), not a chromatic, and even not an edge elimination invariant. Finally, we study whether the Harary polynomials are definable in monadic second-order Logic. Keywords: Generalized colorings; Graph polynomials; Courcelle's Theorem 2020 Mathematics Subject Classification: 05; 05C30; 05C31 1. -
Closest 4-Leaf Power Is Fixed-Parameter Tractable$
Discrete Applied Mathematics 156 (2008) 3345–3361 www.elsevier.com/locate/dam Closest 4-leaf power is fixed-parameter tractableI Michael Dom∗, Jiong Guo, Falk Huffner,¨ Rolf Niedermeier Institut fur¨ Informatik, Friedrich-Schiller-Universitat¨ Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany Received 3 August 2006; received in revised form 18 December 2007; accepted 9 January 2008 Available online 4 March 2008 Abstract The NP-complete CLOSEST 4-LEAF POWER problem asks, given an undirected graph, whether it can be modified by at most r edge insertions or deletions such that it becomes a 4-leaf power. Herein, a 4-leaf power is a graph that can be constructed by considering an unrooted tree—the 4-leaf root—with leaves one-to-one labeled by the graph vertices, where we connect two graph vertices by an edge iff their corresponding leaves are at distance at most 4 in the tree. Complementing previous work on CLOSEST 2-LEAF POWER and CLOSEST 3-LEAF POWER, we give the first algorithmic result for CLOSEST 4-LEAF POWER, showing that CLOSEST 4-LEAF POWER is fixed-parameter tractable with respect to the parameter r. c 2008 Elsevier B.V. All rights reserved. Keywords: Fixed-parameter tractability; Graph algorithm; Graph modification; Graph power; Leaf power; Forbidden subgraph characterization 1. Introduction Graph powers form a classical concept in graph theory, and the rich literature dates back to the sixties of the previous century (see [5, Section 10.6] and [27] for surveys). The k-power of an undirected graph G D .V; E/ is the undirected graph Gk D .V; E0/ with .u; v/ 2 E0 iff there is a path of length at most k between u and v in G. -
Every Monotone Graph Property Is Testable∗
Every Monotone Graph Property is Testable∗ Noga Alon † Asaf Shapira ‡ Abstract A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most well-studied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with one-sided error, and with query complexity depending only on . This result unifies several previous results in the area of property testing, and also implies the testability of well-studied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemer´edi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with one-sided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graph-theory, is that for any monotone graph property P, any graph that is -far from satisfying P, contains a subgraph of size depending on only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is -far from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP ()-far from satisfying one of the properties. -
Graph Properties and Hypergraph Colourings
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 98 (1991) 81-93 81 North-Holland Graph properties and hypergraph colourings Jason I. Brown Department of Mathematics, York University, 4700 Keele Street, Toronto, Ont., Canada M3l lP3 Derek G. Corneil Department of Computer Science, University of Toronto, Toronto, Ont., Canada MSS IA4 Received 20 December 1988 Revised 13 June 1990 Abstract Brown, J.I., D.G. Corneil, Graph properties and hypergraph colourings, Discrete Mathematics 98 (1991) 81-93. Given a graph property P, graph G and integer k 20, a P k-colouring of G is a function Jr:V(G)+ (1,. ) k} such that the subgraph induced by each colour class has property P. When P is closed under induced subgraphs, we can construct a hypergraph HG such that G is P k-colourable iff Hg is k-colourable. This correlation enables us to derive interesting new results in hypergraph chromatic theory from a ‘graphic’ approach. In particular, we build vertex critical hypergraphs that are not edge critical, construct uniquely colourable hypergraphs with few edges and find graph-to-hypergraph transformations that preserve chromatic numbers. 1. Introduction A property P is a collection of graphs (closed under isomorphism) containing K, and K, (all graphs considered here are finite); P is hereditary iff P is closed under induced subgraphs and nontrivial iff P does not contain every graph. Any graph G E P is called a P-graph. Throughout, we shall only be interested in hereditary and nontrivial properties, and P will always denote such a property. -
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) der Fakult¨at f¨ur Informatik und Elektrotechnik der Universit¨at Rostock Vorgelegt von MSc. Ngoc Tuy Nguyen Rostock, November 2009 Gutachter: Prof. Dr. Van Bang Le Universit¨at Rostock. Institut fu¨rInformatik Gutachter: Prof. Dr. Egon Wanke Heinrich- Heine- Universit¨at D¨usseldorf. Institut fu¨rInformatik Gutachter: Prof. Dr. Ekkehard K¨ohler Brandenburgische Technische Universit¨at Cottbus. Institut fu¨rMathematik Tag der ¨offentlichen Verteidigung: 30. Oktober 2009 ii Abstract Given a graph H =(VH , EH ) and a positive integer k, the k-th power of H, written Hk, is the graph obtained from H by adding new edges between any pair of vertices k at distance at most k in H; formally, H =(VH , {xy | 1 ≤ dH (x, y) ≤ k}). A graph G is the k-th power of a graph H if G = Hk, and in this case, H is a k-th root of G. For the cases of k = 2 and k = 3, we say that H2 and H3 is the square, respectively, the cube of H and H is a square root of G = H2, respectively, a cube root of G = H3. In this thesis we study the computational complexity for recognizing k-th pow- ers of general graphs as well as restricted graphs. This work provides new NP- completeness results, good characterizations and efficient algorithms for graph pow- ers. The main results are the following. • There exist reductions proving the NP-completeness for recognizing k-th pow- ers of general graphs for fixed k ≥ 2, recognizing k-th powers of bipartite graphs for fixed k ≥ 3, recognizing k-th powers of chordal graphs, and finding k-th roots of chordal graphs for all fixed k ≥ 2. -
Isomorphism and Embedding Problems for Infinite Limits of Scale
Isomorphism and Embedding Problems for Infinite Limits of Scale-Free Graphs Robert D. Kleinberg ∗ Jon M. Kleinberg y Abstract structure of finite PA graphs; in particular, we give a The study of random graphs has traditionally been characterization of the graphs H for which the expected dominated by the closely-related models (n; m), in number of subgraph embeddings of H in an n-node PA which a graph is sampled from the uniform distributionG graph remains bounded as n goes to infinity. on graphs with n vertices and m edges, and (n; p), in n G 1 Introduction which each of the 2 edges is sampled independently with probability p. Recen tly, however, there has been For decades, the study of random graphs has been dom- considerable interest in alternate random graph models inated by the closely-related models (n; m), in which designed to more closely approximate the properties of a graph is sampled from the uniformG distribution on complex real-world networks such as the Web graph, graphs with n vertices and m edges, and (n; p), in n G the Internet, and large social networks. Two of the most which each of the 2 edges is sampled independently well-studied of these are the closely related \preferential with probability p.The first was introduced by Erd}os attachment" and \copying" models, in which vertices and R´enyi in [16], the second by Gilbert in [19]. While arrive one-by-one in sequence and attach at random in these random graphs have remained a central object \rich-get-richer" fashion to d earlier vertices. -
Recognition of C4-Free and 1/2-Hyperbolic Graphs David Coudert, Guillaume Ducoffe
Recognition of C4-free and 1/2-hyperbolic graphs David Coudert, Guillaume Ducoffe To cite this version: David Coudert, Guillaume Ducoffe. Recognition of C4-free and 1/2-hyperbolic graphs. SIAM Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2014, 28 (3), pp.1601-1617. 10.1137/140954787. hal-01070768 HAL Id: hal-01070768 https://hal.inria.fr/hal-01070768 Submitted on 2 Oct 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ∗ Recognition of C4-free and 1/2-hyperbolic graphs David Coudert†‡ Guillaume Ducoffe†‡§ October 2, 2014 Abstract The shortest-path metric d of a connected graph G is 1/2-hyperbolic if, and only if, it satisfies d(u,v) + d(x,y) ≤ max{d(u,x) + d(v,y), d(u,y) + d(v,x)} + 1, for every 4-tuple u,x,v,y of G. We show that the problem of deciding whether an unweighted graph is 1/2-hyperbolic is subcubic equivalent to the problem of determining whether there is a chordless cycle of length 4 in a graph. An improved algorithm is also given for both problems, taking advantage of fast rectangular matrix multiplication. -
Rainbow Connection and Strong Rainbow Connection Numbers Of
RAINBOW CONNECTION AND STRONG RAINBOW CONNECTION NUMBERS OF Srava Chrisdes Antoro Fakultas Ilmu Komputer, Universitas Gunadarma [email protected] Abstract A rainbow path in an edge coloring of graph is a path in which every two edges are assigned different colors. If a nontrivial connected graph contains a rainbow path for every two vertices in , then is rainbow-connected. The rainbow connection number of is the minimum integer such that is rainbow connnected in an edge- coloring with colors. If a nontrivial connected contains a rainbow path geodesic for every two vertices in , then is strongly rainbow-connected. The strong rainbow connection number of is the minimum integer such that is strongly rainbow-connected in an edge coloring with colors. The rainbow connection and strong rainbow connection number of graph , , and have been found. In this paper, the results will be generalized and the rainbow connection and strong rainbow connection number of graph will be determined. Keywords: rainbow coloring, rainbow connection number, strong rainbow coloring, strong rainbow connection number, graph INTRODUCTION distance between two distinct vertices in . All graphs considered in this paper In 2008, Chartand et. al. introduced the are finite, undirected, and simple. For following two new types of edge labeling. standard terminology and notation, we follow Bondy and Murty (2008). We use Definition 2 (Chartrand et. al., 2008). and to denote the set of Let be a graph and vertices and the set of edges, respectively. , then is called an edge coloring or -coloring on Definition 1 (Prabha and Rajasingh, 2012). where . A path is said to be a The distance between two distinct rainbow path if all edges in that path have vertices and of a connected simple different colors.