DOCSLIB.ORG

Number Theory and Graph Theory Chapter 6

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Number Theory and Graph Theory Chapter 6 1 Number Theory and Graph Theory Chapter 6 Basic concepts and definitions of graph theory By A. Satyanarayana Reddy Department of Mathematics Shiv Nadar University Uttar Pradesh, India E-mail: [email protected] 2 Module-2: Some known graph families and their properties Objectives • Complete graph, null graph, complete bipartite graph • Path graph, cycle graph and wheel graph • Regular graph, Cayley graph and intersection graphs. 1. The simple graph X that has an edge for each pair of vertices is called a complete graph, denoted Kn, where n is the number of vertices of X. 1 3 3 4 2 3 4 5 1 2 1 2 1 2 K2 K3 K4 K5 • There is an edge between any two vertices (any two vertices are adjacent). • It is also called universal graph or clique. n(n−1) • Total number of edges are 2 . • The degree of each vertex is n − 1. • Every simple graph with n vertices is obtained from Kn by deleting some edges. 2. A graph with no edge is called a null graph. We denote null graph with n vertices by Nn 3 3 4 1 2 1 2 1 2 N2 N3 N4 3 3. The cycle graph on n vertices, say 1;2;:::;n, denoted Cn, is a simple graph in which fi; jg is an edge if and only if i − j ≡ ±1 (mod n). 2 2 2 3 3 C4 1 C C3 1 5 1 4 3 4 5 • All the vertices form a cycle. Here cycle we mean, let V = f1;2;3;:::;ng be the vertices of cycle graph Cn, then there is an edge from 1 to 2, then 2 to 3 and so on n − 1 to n and n to 1: • The degree of each vertex is 2. • There are n edges. 4. A graph on n vertices, say 1;2;:::;n, is called a path graph, denoted Pn, if for each i;1 ≤ i ≤ n − 1, the set fi;i + 1g is an edge. 1 2 1 2 3 1 2 3 4 P2 P3 P4 5. A graph X = (V;E) is called bipartite, with bipartition (V1;V2), if V = V1 [V2; V1 \V2 = /0 and each edge of X joins a vertex in V1 with a vertex in V2. For example, in the following graph V = V1 [V2 where V1 = fa;b;c;dg;V2 = fe; f ;g;h;ig. There is no edge between any two vertices of V1. Similarly, there no edge between any two vertices of V2: a b c d e f g h i 4 It is easy to check that (a) every cycle graph of even length is bipartite, and (b) every path graph is bipartite. Whereas, the complete graph Kn, for n ≥ 3 and the cycle graph C2n+1, for n ≥ 1 are not bipartite graphs. 6. A complete bipartite graph is a bipartite graph X with bipartition (V1;V2), in which every vertex in V1 is joined to every vertex in V2: A complete bipartite graph is denoted by Km;n, where jV1j = m and jV2j = n. 1 1 3 1 2 3 K2;2 K1;1 2 K2;1 2 4 • There is an edge between every vertex in V1 to every vertex in V2. • The degree of every vertex in V1 is n whereas in V2, it is m. • The length of every cycle is even. • The number of edges are mn and the number of vertices are m + n. • The graphs K1;m or Km;1 are also called star graphs. 7. Wheel graph: Wheel graph is obtained from a cyclic graph by adding one vertex to it, called the hub, and edges from this vertex to all other vertices. • It is denoted by WHn. • It consists of n + 1 vertices and 2n edges. • The degree of every vertex excluding the hub is 3. For the hub it is n. 5 3 4 3 4 5 1 2 1 2 WH3 WH4 8. A graph (digraph) X is said to be k- regular if d(v) = k (d+(v) = d−(v) = k) for all v 2 V. • The degree of every vertex is k. • If the number of vertices are odd then using the hand-shaking lemma, its regularity cannot be odd. • Complete graph is n − 1 regular. • Complete bipartite graph is regular graph if m = n. • Cyclic graph is 2-regular. • Path is regular if it has 2 vertices. • Wheel graph is regular only when n = 3: • The Johnson graphs (see Example 2.1 in module 1 of chapter 6) and k-cube graphs (see Example 2.3 in module 1 of chapter 6) are regular. • The Petersen graph (see Example 2.1 in module 1 of chapter 6) is 3-regular. 9. Let G be a group and let S be a non-empty subset of G that does not contain the identity element of G. Then the Cayley digraph/graph associated with the pair (G;S), denoted Cay(G;S), has the set G as its vertex set and for any two vertices x;y 2 G; (x;y) is an edge if xy−1 2 S. Observe that Cay(G;S) is a graph if and only if S is closed with respect to inverse (S = S−1 = fs−1 : s 2 Sg). Also, the graph is k-regular if S has k elements. The set S is called the 6 connection set of the graph. For example, G = S3 = fe;(12);(13);(23);(123);(132)g be the symmetric group on 3 symbols and S = f(12);(13);(23)g. Then, S is a generating set of G as every permutation can be expressed as product of transpositions. The corresponding Cayley graph Cay(G;S) is K3;3, as shown below. e (12) (13) (123) (132) (23) Cayley graphs on cyclic groups are called circulant graphs. An important class of circulant graphs are Payley graphs. We define them in the following manner. Let q be a prime power such that q ≡ 1 (mod 4), and Fq denote a finite field of order q, the Payley graph P(q) of order q is a graph whose vertices are elements of the finite field Fq in which two vertices are adjacent if and only if their difference is a non-zero square in Fq: Since q−1 q−1 in Fq there are 2 non-zero squares (quadratic residues) hence P(q) is 2 -regular graph. As we aware −1 is a square in modulo q if and only if q ≡ 1 (mod 4). Consequently a − b is a square if and only if b − a is a square. To know more about Payley graphs refer Algebraic graph theory by Chris Godsil and Gordan Royal. On the other hand if q ≡ 3 (mod 4), then P(q) is a digraph. The following graph is P(13): 7 3 2 1 4 0 5 12 6 11 7 10 8 9 10. Let S1;S2;:::;Sn be non-empty subsets of a set S. Then we define the intersection graph, overlap graph, containment graph and disjointedness graphs as follows. The vertex set of all these graphs is V = fS1;S2;:::;Sng. There is an edge between Si and S j for i 6= j if Intersection graph Si \ S j 6= /0: Overlap graph Si \ S j 6= /0: and Si * S j and S j * Si Containment graph Si ⊂ S j or S j ⊂ Si: Disjointedness graph Si \ S j = /0: Depending on the nature or the geometric configurations of the sets S1;S2;:::;Sn, many im- portant subclasses of intersection graphs were generated and studied. Few of them are interval graphs, tolerance graphs, chordal graphs, unit disk graphs, line graphs, circular arc graphs etc. For example, a graph is an interval graph if it can be represented as intersection graph of intervals. Here S1;S2;:::;Sn are intervals. Following figure illustrates the construction of interval graph with five intervals. 8 5 5 4 4 3 3 1 2 1 2 Many problems which are difficult to solve for general graphs are easy to solve once we confine to interval graphs. Further interval graphs have a number of applications in different areas, for example, archeology, scheduling, biology etc. Few Solved Problems 1. What is the smallest number n such that the complete graph Kn has at least 500 edges? n(n−1) n(n−1) Solution:Since number of edges in Kn is 2 , hence 2 ≥ 500. Or equivalently n(n − 1) ≥ 1000: Thus smallest value of n is 33: n2 2. Prove that the number of edges in a bipartite graph with n vertices is at most 4 . Solution:Let X = (V1 [V2;E) be a bipartite graph with n vertices. Let jV1j = x, then jV2j = n − x. By definition, X has at most f (x) = x(n − x) edges. Since f 0(x) = n − 2x and f 00 = −2, n n2 consequently f (x) has its maximum value at x = 2 and the value is 4 : 3. Let X be a k-regular graph, where k is an odd number. Then prove that number of edges in X is a multiple of k. Solution:We have 2jEj = kjVj. Hence k divides 2jEj but gcd(2;k) = 1 hence k divides jEj. 4. For what values of m and n, the complete bipartite graph Km;n has more number of vertices than edges. Solution:We know that the number of vertices and edges in Km;n are m+n and mn, respectively.
Load more

Recommended publications
  • On Treewidth and Graph Minors
    On Treewidth and Graph Minors Daniel John Harvey Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy February 2014 Department of Mathematics and Statistics The University of Melbourne Produced on archival quality paper ii Abstract Both treewidth and the Hadwiger number are key graph parameters in structural and al- gorithmic graph theory, especially in the theory of graph minors. For example, treewidth demarcates the two major cases of the Robertson and Seymour proof of Wagner's Con- jecture. Also, the Hadwiger number is the key measure of the structural complexity of a graph. In this thesis, we shall investigate these parameters on some interesting classes of graphs. The treewidth of a graph defines, in some sense, how \tree-like" the graph is. Treewidth is a key parameter in the algorithmic field of fixed-parameter tractability. In particular, on classes of bounded treewidth, certain NP-Hard problems can be solved in polynomial time. In structural graph theory, treewidth is of key interest due to its part in the stronger form of Robertson and Seymour's Graph Minor Structure Theorem. A key fact is that the treewidth of a graph is tied to the size of its largest grid minor. In fact, treewidth is tied to a large number of other graph structural parameters, which this thesis thoroughly investigates. In doing so, some of the tying functions between these results are improved. This thesis also determines exactly the treewidth of the line graph of a complete graph. This is a critical example in a recent paper of Marx, and improves on a recent result by Grohe and Marx.
    [Show full text]
  • Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse
    Exclusive Graph Searching Lélia Blin, Janna Burman, Nicolas Nisse To cite this version: Lélia Blin, Janna Burman, Nicolas Nisse. Exclusive Graph Searching. Algorithmica, Springer Verlag, 2017, 77 (3), pp.942-969. 10.1007/s00453-016-0124-0. hal-01266492 HAL Id: hal-01266492 https://hal.archives-ouvertes.fr/hal-01266492 Submitted on 2 Feb 2016 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. Exclusive Graph Searching∗ L´eliaBlin Sorbonne Universit´es, UPMC Univ Paris 06, CNRS, Universit´ed'Evry-Val-d'Essonne. LIP6 UMR 7606, 4 place Jussieu 75005, Paris, France [email protected] Janna Burman LRI, Universit´eParis Sud, CNRS, UMR-8623, France. [email protected] Nicolas Nisse Inria, France. Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, Sophia Antipolis, France. [email protected] February 2, 2016 Abstract This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. This problem has been mainly studied for its relationship with the pathwidth of graphs. All variants of this problem assume that any node can be simultaneously occupied by several searchers.
    [Show full text]
  • Blast Domination for Mycielski's Graph of Graphs
    International Journal of Engineering and Advanced Technology (IJEAT) ISSN: 2249 – 8958, Volume-8, Issue-6S3, September 2019 Blast Domination for Mycielski’s Graph of Graphs K. Ameenal Bibi, P.Rajakumari AbstractThe hub of this article is a search on the behavior of is the minimum cardinality of a distance-2 dominating set in the Blast domination and Blast distance-2 domination for 퐺. Mycielski’s graph of some particular graphs and zero divisor graphs. Definition 2.3[7] A non-empty subset 퐷 of 푉 of a connected graph 퐺 is Key Words:Blast domination number, Blast distance-2 called a Blast dominating set, if 퐷 is aconnected dominating domination number, Mycielski’sgraph. set and the induced sub graph < 푉 − 퐷 >is triple connected. The minimum cardinality taken over all such Blast I. INTRODUCTION dominating sets is called the Blast domination number of The concept of triple connected graphs was introduced by 퐺and is denoted by tc . Paulraj Joseph et.al [9]. A graph is said to be triple c (G) connected if any three vertices lie on a path in G. In [6] the Definition2.4 authors introduced triple connected domination number of a graph. A subset D of V of a nontrivial graph G is said to A non-empty subset 퐷 of vertices in a graph 퐺 is a blast betriple connected dominating set, if D is a dominating set distance-2 dominating set if every vertex in 푉 − 퐷 is within and <D> is triple connected. The minimum cardinality taken distance-2 of atleast one vertex in 퐷.
    [Show full text]
  • Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth K
    Empirical Comparison of Greedy Strategies for Learning Markov Networks of Treewidth k K. Nunez, J. Chen and P. Chen G. Ding and R. F. Lax B. Marx Dept. of Computer Science Dept. of Mathematics Dept. of Experimental Statistics LSU LSU LSU Baton Rouge, LA 70803 Baton Rouge, LA 70803 Baton Rouge, LA 70803 Abstract—We recently proposed the Edgewise Greedy Algo- simply one less than the maximum of the orders of the rithm (EGA) for learning a decomposable Markov network of maximal cliques [5], a treewidth k DMN approximation to treewidth k approximating a given joint probability distribution a joint probability distribution of n random variables offers of n discrete random variables. The main ingredient of our algorithm is the stepwise forward selection algorithm (FSA) due computational benefits as the approximation uses products of to Deshpande, Garofalakis, and Jordan. EGA is an efficient alter- marginal distributions each involving at most k + 1 of the native to the algorithm (HGA) by Malvestuto, which constructs random variables. a model of treewidth k by selecting hyperedges of order k +1. In The treewidth k DNM learning problem deals with a given this paper, we present results of empirical studies that compare empirical distribution P ∗ (which could be given in the form HGA, EGA and FSA-K which is a straightforward application of FSA, in terms of approximation accuracy (measured by of a distribution table or in the form of a set of training data), KL-divergence) and computational time. Our experiments show and the objective is to find a DNM (i.e., a chordal graph) that (1) on the average, all three algorithms produce similar G of treewidth k which defines a probability distribution Pµ approximation accuracy; (2) EGA produces comparable or better that ”best” approximates P ∗ according to some criterion.
    [Show full text]
  • Determination and Testing the Domination Numbers of Tadpole Graph, Book Graph and Stacked Book Graph Using MATLAB Ayhan A
    College of Basic Education Researchers Journal Vol. 10, No. 1 Determination and Testing the Domination Numbers of Tadpole Graph, Book Graph and Stacked Book Graph Using MATLAB Ayhan A. khalil Omar A.Khalil Technical College of Mosul -Foundation of Technical Education Received: 11/4/2010 ; Accepted: 24/6/2010 Abstract: A set is said to be dominating set of G if for every there exists a vertex such that . The minimum cardinality of vertices among dominating set of G is called the domination number of G denoted by . We investigate the domination number of Tadpole graph, Book graph and Stacked Book graph. Also we test our theoretical results in computer by introducing a matlab procedure to find the domination number , dominating set S and draw this graph that illustrates the vertices of domination this graphs. It is proved that: . Keywords: Dominating set, Domination number, Tadpole graph, Book graph, stacked graph. إﯾﺟﺎد واﺧﺗﺑﺎر اﻟﻌدد اﻟﻣﮭﯾﻣن(اﻟﻣﺳﯾطر)Domination number ﻟﻠﺑﯾﺎﻧﺎت ﺗﺎدﺑول (Tadpole graph)، ﺑﯾﺎن ﺑوك (Book graph) وﺑﯾﺎن ﺳﺗﺎﻛﯾت ﺑوك (Stacked Book graph) ﺑﺎﺳﺗﻌﻣﺎل اﻟﻣﺎﺗﻼب أﯾﮭﺎن اﺣﻣد ﺧﻠﯾل ﻋﻣر اﺣﻣد ﺧﻠﯾل اﻟﻛﻠﯾﺔ اﻟﺗﻘﻧﯾﺔ-ﻣوﺻل- ھﯾﺋﺔ اﻟﺗﻌﻠﯾم اﻟﺗﻘﻧﻲ ﻣﻠﺧص اﻟﺑﺣث : ﯾﻘﺎل ﻷﯾﺔ ﻣﺟﻣوﻋﺔ ﺟزﺋﯾﺔ S ﻣن ﻣﺟﻣوﻋﺔ اﻟرؤوس V ﻓﻲ ﺑﯾﺎن G ﺑﺄﻧﻬﺎ ﻣﺟﻣوﻋﺔ ﻣﻬﯾﻣﻧﺔ Dominating set إذا ﻛﺎن ﻟﻛل أرس v ﻓﻲ اﻟﻣﺟﻣوﻋﺔ V-S ﯾوﺟد أرس u ﻓﻲS ﺑﺣﯾث uv ﺿﻣن 491 Ayhan A. & Omar A. ﺣﺎﻓﺎت اﻟﺑﯾﺎن،. وﯾﻌرف اﻟﻌدد اﻟﻣﻬﯾﻣن Domination number ﺑﺄﻧﻪ أﺻﻐر ﻣﺟﻣوﻋﺔ أﺳﺎﺳﯾﺔ ﻣﻬﯾﻣﻧﺔ Domination. ﻓﻲ ﻫذا اﻟﺑﺣث ﺳوف ﻧدرس اﻟﻌدد اﻟﻣﻬﯾﻣن Domination number ﻟﺑﯾﺎن ﺗﺎدﺑول (Tadpole graph) ، ﺑﯾﺎن ﺑوك (Book graph) وﺑﯾﺎن ﺳﺗﺎﻛﯾت ﺑوك ( Stacked Book graph).
    [Show full text]
  • Discrete Mathematics Rooted Directed Path Graphs Are Leaf Powers
    Discrete Mathematics 310 (2010) 897–910 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Rooted directed path graphs are leaf powers Andreas Brandstädt a, Christian Hundt a, Federico Mancini b, Peter Wagner a a Institut für Informatik, Universität Rostock, D-18051, Germany b Department of Informatics, University of Bergen, N-5020 Bergen, Norway article info a b s t r a c t Article history: Leaf powers are a graph class which has been introduced to model the problem of Received 11 March 2009 reconstructing phylogenetic trees. A graph G D .V ; E/ is called k-leaf power if it admits a Received in revised form 2 September 2009 k-leaf root, i.e., a tree T with leaves V such that uv is an edge in G if and only if the distance Accepted 13 October 2009 between u and v in T is at most k. Moroever, a graph is simply called leaf power if it is a Available online 30 October 2009 k-leaf power for some k 2 N. This paper characterizes leaf powers in terms of their relation to several other known graph classes. It also addresses the problem of deciding whether a Keywords: given graph is a k-leaf power. Leaf powers Leaf roots We show that the class of leaf powers coincides with fixed tolerance NeST graphs, a Graph class inclusions well-known graph class with absolutely different motivations. After this, we provide the Strongly chordal graphs largest currently known proper subclass of leaf powers, i.e, the class of rooted directed path Fixed tolerance NeST graphs graphs.
    [Show full text]
  • Constructing Arbitrarily Large Graphs with a Specified Number Of
    Electronic Journal of Graph Theory and Applications 4 (1) (2019), 1–8 Constructing Arbitrarily Large Graphs with a Specified Number of Hamiltonian Cycles Michael Haythorpea aSchool of Computer Science, Engineering and Mathematics, Flinders University, 1284 South Road, Clovelly Park, SA 5042, Australia michael.haythorpe@flinders.edu.au Abstract A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing 5n − 9 vertices and k Hamiltonian cycles for any choice of integers n ≥ k ≥ 4. The construction is not designed to be minimal in any sense, but rather to ensure that the graphs produced remain non-trivial instances of the Hamiltonian cycle problem even when k is chosen to be much smaller than n. Keywords: Hamiltonian cycles, Graph Construction, Broken Crown Mathematics Subject Classification : 05C45 1. Introduction The Hamiltonian cycle problem (HCP) is a famous NP-complete problem in which one must determine whether a given graph contains a simple cycle traversing all vertices of the graph, or not. Such a simple cycle is called a Hamiltonian cycle (HC), and a graph containing at least one arXiv:1902.10351v1 [math.CO] 27 Feb 2019 Hamiltonian cycle is said to be a Hamiltonian graph. Typically, randomly generated graphs (such as Erdos-R˝ enyi´ graphs), if connected, are Hamil- tonian and contain many Hamiltonian cycles. Although HCP is an NP-complete problem, for these graphs it is often fairly easy for a sophisticated heuristic (e.g. see Concorde [1], Keld Helsgaun’s LKH [4] or Snakes-and-ladders Heuristic [2]) to discover one of the multitude of Hamiltonian cy- cles through a clever search.
    [Show full text]
  • Dominating Cycles in Halin Graphs*
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 86 (1990) 215-224 215 North-Holland DOMINATING CYCLES IN HALIN GRAPHS* Mirosiawa SKOWRONSKA Institute of Mathematics, Copernicus University, Chopina 12/18, 87-100 Torun’, Poland Maciej M. SYStO Institute of Computer Science, University of Wroclaw, Przesmyckiego 20, 51-151 Wroclaw, Poland Received 2 December 1988 A cycle in a graph is dominating if every vertex lies at distance at most one from the cycle and a cycle is D-cycle if every edge is incident with a vertex of the cycle. In this paper, first we provide recursive formulae for finding a shortest dominating cycle in a Hahn graph; minor modifications can give formulae for finding a shortest D-cycle. Then, dominating cycles and D-cycles in a Halin graph H are characterized in terms of the cycle graph, the intersection graph of the faces of H. 1. Preliminaries The various domination problems have been extensively studied. Among them is the problem whether a graph has a dominating cycle. All graphs in this paper have no loops and multiple edges. A dominating cycle in a graph G = (V(G), E(G)) is a subgraph C of G which is a cycle and every vertex of V(G) \ V(C) is adjacent to a vertex of C. There are graphs which have no dominating cycles, and moreover, determining whether a graph has a dominating cycle on at most 1 vertices is NP-complete even in the class of planar graphs [7], chordal, bipartite and split graphs [3].
    [Show full text]
  • Hamiltonian Cycles in Generalized Petersen Graphs Let N and K Be
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY, Series B 24, 181-188 (1978) Hamiltonian Cycles in Generalized Petersen Graphs Kozo BANNAI Information Processing Research Center, Central Research Institute of Electric Power Industry, 1-6-I Ohternachi, Chiyoda-ku, Tokyo, Japan Communicated by the Managing Edirors Received March 5, 1974 Watkins (J. Combinatorial Theory 6 (1969), 152-164) introduced the concept of generalized Petersen graphs and conjectured that all but the original Petersen graph have a Tait coloring. Castagna and Prins (Pacific J. Math. 40 (1972), 53-58) showed that the conjecture was true and conjectured that generalized Petersen graphs G(n, k) are Hamiltonian unless isomorphic to G(n, 2) where n E S(mod 6). The purpose of this paper is to prove the conjecture of Castagna and Prins in the case of coprime numbers n and k. 1. INTRODUCTION Let n and k be coprime positive integers with k < n, i.e., (n, k) = 1. The generalized Petersengraph G(n, k) consistsof 2n vertices (0, 1, 2,..., n - 1; O’, l’,..., n - I’} and 3n edgesof the form (i, i + l), (i’, i + l’), (i, ik’) for 0 < i < n - 1, where all numbers are read modulo n. The set of edges ((i, i + 1)) and the set of edges((i’, i f l’)} are called the outer rim and the inner rim, respectively, and edges{(i, ik’)) are called spokes. Note that the above definition of generalized Petersen graphs differs from that of Watkins /4] and Castagna and Prins [2] in the numbering of the vertices.
    [Show full text]
  • Compact Representation of Graphs with Small Bandwidth and Treedepth
    Compact Representation of Graphs with Small Bandwidth and Treedepth Shahin Kamali University of Manitoba Winnipeg, MB, R3T 2N2, Canada [email protected] Abstract We consider the problem of compact representation of graphs with small bandwidth as well as graphs with small treedepth. These parameters capture structural properties of graphs that come in useful in certain applications. We present simple navigation oracles that support degree and adjacency queries in constant time and neighborhood query in constant time per neighbor. For graphs of bandwidth k, our oracle takesp (k + dlog 2ke)n + o(kn) bits. By way of an enumeration technique, we show that (k − 5 k − 4)n − O(k2) bits are required to encode a graph of bandwidth k and size n. For graphs of treedepth k, our oracle takes (k + dlog ke + 2)n + o(kn) bits. We present a lower bound that certifies our oracle is succinct for certain values of k 2 o(n). 1 Introduction Graphs are among the most relevant ways to model relationships between objects. Given the ever-growing number of objects to model, it is increasingly important to store the underlying graphs in a compact manner in order to facilitate designing efficient algorithmic solutions. One simple way to represent graphs is to consider them as random objects without any particular structure. There are two issues, however, with such general approach. First, many computational problems are NP-hard on general graphs and often remain hard to approximate. Second, random graphs are highly incompressible and cannot be stored compactly [1]. At the same time, graphs that arise in practice often have combinatorial structures that can -and should- be exploited to provide space-efficient representations.
    [Show full text]
  • Branch-Depth: Generalizing Tree-Depth of Graphs
    Branch-depth: Generalizing tree-depth of graphs ∗1 †‡23 34 Matt DeVos , O-joung Kwon , and Sang-il Oum† 1Department of Mathematics, Simon Fraser University, Burnaby, Canada 2Department of Mathematics, Incheon National University, Incheon, Korea 3Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, Korea 4Department of Mathematical Sciences, KAIST, Daejeon, Korea [email protected], [email protected], [email protected] November 5, 2020 Abstract We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the no- tions of tree-depth and shrub-depth of graphs as follows. For a graph G = (V, E) and a subset A of E we let λG(A) be the number of vertices incident with an edge in A and an edge in E A. For a subset X of V , \ let ρG(X) be the rank of the adjacency matrix between X and V X over the binary field. We prove that a class of graphs has bounded\ tree-depth if and only if the corresponding class of functions λG has arXiv:1903.11988v2 [math.CO] 4 Nov 2020 bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions ρG has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree- depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by restriction.
    [Show full text]
  • Graph in Data Structure with Example
    Graph In Data Structure With Example Tremain remove punctually while Memphite Gerald valuate metabolically or soaks affrontingly. Grisliest Sasha usually reconverts some singlesticks or incage historiographically. If innocent or dignified Verge usually cleeking his pewit clipped skilfully or infests infra and exchangeably, how centum is Rad? What is integer at which means that was merely an organizational level overview of. Removes the specified node. Particularly find and examples. What facial data structure means. Graphs Murray State University. For example with edge going to structure of both have? Graphs in Data Structure Tutorial Ride. That instead of structure and with example as being compared. The traveling salesman problem near a folder example of using a tree algorithm to. When at the neighbors of efficient current node are considered, it marks the current node as visited and is removed from the unvisited list. You with example. There consider no isolated nodes in connected graph. The data structures in a stack of linked lists are featured in data in java some definitions that these files if a direction? We can be incident with example data structures! Vi and examples will only be it can be a structure in a source. What are examples can be identified by edges with. All data structures? A rescue Study a Graph Data Structure ijarcce. We can say that there are very much in its length of another and put a vertical. The edge uv, which functions that, we can be its direct support this essentially means. If they appear in data structures and solve other values for our official venues for others with them is another.
    [Show full text]