DOCSLIB.ORG

The Subchromatic Index of Graphs

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Subchromatic Index of Graphs t m Tatra Mt Math Publ Mathematical Publications THE SUBCHROMATIC INDEX OF GRAPHS Jir Fiala Van Bang Le ABSTRACT In an edge coloring of a graph each color class forms a subgraph without a path of length twoamatching An edge sub coloring extends this concept each color class in an edge sub coloring forms a subgraph without path of length three While every graph with maximum degree at most twoisedge sub colorable wepoint out in this pap er that recognizing edge sub colorable graphs with maximum degree three is NPcomplete even when restricted to trianglefree graphs As bypro ducts we obtain NP completeness results for the star index and the sub chromatic numb er for several classes of graphs It is also proved that recognizing edge sub colorable graphs is NP complete Moreover edge sub colorings and sub chromatic index of various basic graph classes are studied In particular we showthatevery unicyclic graph is edge sub colorable and edge sub colorable unicyclic graphs have a simple structure allowing an easy linear time recognition Intro duction Let G V E b e a graph An independent set a clique is a set of pairwise nonadjacent adjacent vertices For W V the subgraph of G induced by W is denoted by GW For F E the symbol V F denotes the set of endvertices of edges from F and GF V F F is the subgraph of G induced by the edge set F A prop er vertex r coloring of G is a partition V V into disjoint in r dep endent sets called color classes of the coloring The chromatic number G is the smallest number r for which G admits a vertex r coloring One of the most interesting generalizations of the classical vertex coloring is the notion of vertex sub coloring see A vertex r subcoloring is a partition V V of V where each color class V consists of disjoint cliques of various r i sizes The smallest number r for which G has a vertex r sub coloring is called the subchromatic number GofG s M a thematics Subject Classification C C Keywordscomputationalcomplexity line graphs sub coloring sub chromatic index Supp orted by the Ministry of Education of the Czech Republic as pro ject M JIRI FIALA VAN BANG LE Note that a partition V V of V isavertex r coloring of G V Eif r and only if for each i the graph GV do es not contain a P as an induced i subgraph The partition is a vertex r sub coloring if and only if for each i the graph GV do es not contain a P as an induced subgraph P denotes the i k path on k vertices A prop er edge r coloring is a partition E E of E into color classes r E in whichevery two distinct edges do not have an endvertex in common ie i each E forms a matching The chromatic index G is the smallest number i r for which G admits an edge r coloring Clearly a partition E E of E r is an edge r coloring of G V E if and only if for each i the subgraph GE i do es not contain a P as a not necessarily induced subgraph This observation leads to the following natural generalization of the classical edge coloring Definition An edge r subcoloring of the edges of a graph G V Eis a partition E E of E into disjoint color classes E such that for each E r i i the graph GE contains no P as a not necessarily induced subgraph The i subchromatic index G is the smallest number r for which G admits an edge s r sub coloring Remark Obviously a partition E E of E G is an edge r sub coloring r of G if and only if for each i the connected comp onents of GE are stars or i triangles where a star is a complete bipartite graph K for some s s A related notion that has b een studied in the literature is as follows A partition E E of E Gisastar partition of G if for each i the connected r comp onents of GE are stars The star index GofG is the smallest i number r for which G has a star partition into r subsets E cf i Clearly G G for all graphs G and it holds that G G s s whenever G is trianglefree Recall that the line graph LG of a graph G has the edges of G as vertices and two distinct edges e e are adjacentin LG whenever they have an endver tex in common It is wellknown that prop er edge colorings of G corresp ond to prop er vertex colorings of LG and vice versa In particular G LG Likewise the following fact is easy to see Fact Edge sub colorings of a graph G corresp ond to vertex sub colorings of the line graph LG of G and vice versa In particular G LG s s Our terminology of edge sub coloring is intended to recall this fact THE SUBCHROMATIC INDEX OF GRAPHS Basic prop erties and examples The sub chromatic index of a disconnected graph is the maximum sub chro matic index among those of its connected comp onents Hence without loss of generalitywe assume throughout this pap er that all graphs are connected By the denition the edge sub chromatic index is monotone with resp ect to graph inclusion ie G G G G The next observation shows a s s close link b etween sub chromatic indices of graphs where one is formed from the other by removal of a vertex Observation For any graph G and anyvertex v of G it holds that G G n v s s Proo f Any sub coloring of G n v can b e extended to a sub coloring of G by using an extra new color on all edges incidentwith v General lower and upp er b ounds for the sub chromatic index are given b elow Let G b e the maximum degree of a vertex in the graph G Proposition Any graph G with m edges on n vertices satises m G G s n m Moreover G if G is trianglefree s n P r o o f Since every color class consists of stars and triangles it may contain at most n edges In the sub coloring each edge has to b e colored and the lower b ound follows Note that a color class in a graph on n vertices can have n edges if and only if n isamultiple of and the class itself is a covering of the vertices by n disjoint triangles In trianglefree graphs no such class exists and the lower b ound can b e made sharp How to obtain the upp er b ound wehavefromFact LG G s s l m l m LG G G where the inequality for the sub chro matic number was shown in To nd a valid sub coloring using at most G edge colors ecientlywe may pro ceed greedily on the vertex set with each new vertex u assign colors to its adjacent edges as follows for an edge u v pick a color that is not used on an already colored edge incident with v Such a sub coloring is trianglefree and all stars have the prop erty that the center of the star is the latest vertex of the star in the order JIRI FIALA VAN BANG LE C Observe that the upp er b ound is attained eg for the cycle s C or the Petersen graph P P The last prop erty follows s for any cubic graph whichcontains C as an induced subgraph it is imp ossible to extend a valid sub coloring to all edges incident to the cycle C See also Figure r G r Moreover Corollary Any r regular graph G satises s r G if G is trianglefree s Trees and cycles For trees and cycles the sub chromatic index can b e determined explicitly as follows Proposition i For anytree T T T T if and only if T is not s s a star ii C and C C for all n s n n s P r o o f Color greedily Cacti A cactus is a connected graph in whichevery blo ck maximal connected sub graph is an edge or a cycle Equivalently a graph G is a cactus if and only if every two cycles in G are edgedisjoint Proposition For all cacti G wehave G G Moreover an s edge sub coloring can b e found in linear time Proo f Let T b e a breadthrst search bfs tree of G ro oted at vertex v We claim that all edges of G outside T form a matching During the searchwe arrange vertices into levels based on the distance from the initial vertex Since G is a cactus the tree T misses from each o dd cycle the edge connecting the twovertices at the highest level Similarly for an even cycle one of the two edges incident with the vertex at the highest level remains outside T Now observe the given twoedges e e of E G n E T either e is separated from e bythelowest vertex of the cycle containing e or viceversa We color T with two colors and use the matching E G n E T as the third color This shows G Since a bfstree can b e p erformed in linear time Prop osition follows THE SUBCHROMATIC INDEX OF GRAPHS Weleave it as an op en problem whether cacti with sub chromatic index at most allow a simple structural description Since all cacti have treewidth up p erb ounded bytheirsubchromatic index can b e computed in p olynomial time as we will show later in Section Observe that in any edge sub colored cactus G eachvertex of degree at least is either a center of a mono chromatic star or it b elongs to a mono chro matic triangle Let us direct the edges of mono chromatic stars K k k towards their centers The other edges remain undirected Clearlynovertex of degree at least in G is of outdegree or more Each directed cycle is of even length since the colors of stars must alternate Figure shows some cacti G that do allowsuch orientation of their cycles G and hence have s C k k Figure Some cacti with sub chromatic index Unicycli c graphs A connected graph is unicyclic if it contains exactly one cycle As we show now all edge sub colorable unicyclic graphs have a simple structure and hence can b e easily recognized in linear time Theorem For any unicyclic graph G wehave G Moreover s G if and only if the only cycle C of G has length k k s and all vertices of C are of degree at least in G P r o o f Since unicyclic graphs are cacti the rst part follows from Prop o sition Wehave shown ab ove that if an o dd cycle C k contains no k vertex of degree
Load more

Recommended publications
  • Structural Parameterizations of Clique Coloring
    Structural Parameterizations of Clique Coloring Lars Jaffke University of Bergen, Norway lars.jaff[email protected] Paloma T. Lima University of Bergen, Norway [email protected] Geevarghese Philip Chennai Mathematical Institute, India UMI ReLaX, Chennai, India [email protected] Abstract A clique coloring of a graph is an assignment of colors to its vertices such that no maximal clique is monochromatic. We initiate the study of structural parameterizations of the Clique Coloring problem which asks whether a given graph has a clique coloring with q colors. For fixed q ≥ 2, we give an O?(qtw)-time algorithm when the input graph is given together with one of its tree decompositions of width tw. We complement this result with a matching lower bound under the Strong Exponential Time Hypothesis. We furthermore show that (when the number of colors is unbounded) Clique Coloring is XP parameterized by clique-width. 2012 ACM Subject Classification Mathematics of computing → Graph coloring Keywords and phrases clique coloring, treewidth, clique-width, structural parameterization, Strong Exponential Time Hypothesis Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.49 Related Version A full version of this paper is available at https://arxiv.org/abs/2005.04733. Funding Lars Jaffke: Supported by the Trond Mohn Foundation (TMS). Acknowledgements The work was partially done while L. J. and P. T. L. were visiting Chennai Mathematical Institute. 1 Introduction Vertex coloring problems are central in algorithmic graph theory, and appear in many variants. One of these is Clique Coloring, which given a graph G and an integer k asks whether G has a clique coloring with k colors, i.e.
    [Show full text]
  • Clique-Width and Tree-Width of Sparse Graphs
    Clique-width and tree-width of sparse graphs Bruno Courcelle Labri, CNRS and Bordeaux University∗ 33405 Talence, France email: [email protected] June 10, 2015 Abstract Tree-width and clique-width are two important graph complexity mea- sures that serve as parameters in many fixed-parameter tractable (FPT) algorithms. The same classes of sparse graphs, in particular of planar graphs and of graphs of bounded degree have bounded tree-width and bounded clique-width. We prove that, for sparse graphs, clique-width is polynomially bounded in terms of tree-width. For planar and incidence graphs, we establish linear bounds. Our motivation is the construction of FPT algorithms based on automata that take as input the algebraic terms denoting graphs of bounded clique-width. These algorithms can check properties of graphs of bounded tree-width expressed by monadic second- order formulas written with edge set quantifications. We give an algorithm that transforms tree-decompositions into clique-width terms that match the proved upper-bounds. keywords: tree-width; clique-width; graph de- composition; sparse graph; incidence graph Introduction Tree-width and clique-width are important graph complexity measures that oc- cur as parameters in many fixed-parameter tractable (FPT) algorithms [7, 9, 11, 12, 14]. They are also important for the theory of graph structuring and for the description of graph classes by forbidden subgraphs, minors and vertex-minors. Both notions are based on certain hierarchical graph decompositions, and the associated FPT algorithms use dynamic programming on these decompositions. To be usable, they need input graphs of "small" tree-width or clique-width, that furthermore are given with the relevant decompositions.
    [Show full text]
  • Parameterized Algorithms for Recognizing Monopolar and 2
    Parameterized Algorithms for Recognizing Monopolar and 2-Subcolorable Graphs∗ Iyad Kanj1, Christian Komusiewicz2, Manuel Sorge3, and Erik Jan van Leeuwen4 1School of Computing, DePaul University, Chicago, USA, [email protected] 2Institut für Informatik, Friedrich-Schiller-Universität Jena, Germany, [email protected] 3Institut für Softwaretechnik und Theoretische Informatik, TU Berlin, Germany, [email protected] 4Max-Planck-Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany, [email protected] October 16, 2018 Abstract A graph G is a (ΠA, ΠB)-graph if V (G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA, ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA, ΠB )-graph. There are many (ΠA, ΠB)-Recognition problems, in- cluding the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA, ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA, ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameter- ized by the number of maximal cliques in G[A]. We complement our algo- rithmic results with several hardness results for (ΠA, ΠB )-Recognition. arXiv:1702.04322v2 [cs.CC] 4 Jan 2018 1 Introduction A (ΠA, ΠB)-graph, for graph properties ΠA, ΠB , is a graph G = (V, E) for which V admits a partition into two sets A, B such that G[A] satisfies ΠA and G[B] satisfies ΠB. There is an abundance of (ΠA, ΠB )-graph classes, and important ones include bipartite graphs (which admit a partition into two independent ∗A preliminary version of this paper appeared in SWAT 2016, volume 53 of LIPIcs, pages 14:1–14:14.
    [Show full text]
  • Sub-Coloring and Hypo-Coloring Interval Graphs⋆
    Sub-coloring and Hypo-coloring Interval Graphs? Rajiv Gandhi1, Bradford Greening, Jr.1, Sriram Pemmaraju2, and Rajiv Raman3 1 Department of Computer Science, Rutgers University-Camden, Camden, NJ 08102. E-mail: [email protected]. 2 Department of Computer Science, University of Iowa, Iowa City, Iowa 52242. E-mail: [email protected]. 3 Max-Planck Institute for Informatik, Saarbr¨ucken, Germany. E-mail: [email protected]. Abstract. In this paper, we study the sub-coloring and hypo-coloring problems on interval graphs. These problems have applications in job scheduling and distributed computing and can be used as “subroutines” for other combinatorial optimization problems. In the sub-coloring problem, given a graph G, we want to partition the vertices of G into minimum number of sub-color classes, where each sub-color class induces a union of disjoint cliques in G. In the hypo-coloring problem, given a graph G, and integral weights on vertices, we want to find a partition of the vertices of G into sub-color classes such that the sum of the weights of the heaviest cliques in each sub-color class is minimized. We present a “forbidden subgraph” characterization of graphs with sub-chromatic number k and use this to derive a a 3-approximation algorithm for sub-coloring interval graphs. For the hypo-coloring problem on interval graphs, we first show that it is NP-complete and then via reduction to the max-coloring problem, show how to obtain an O(log n)-approximation algorithm for it. 1 Introduction Given a graph G = (V, E), a k-sub-coloring of G is a partition of V into sub-color classes V1,V2,...,Vk; a subset Vi ⊆ V is called a sub-color class if it induces a union of disjoint cliques in G.
    [Show full text]
  • Shrub-Depth a Successful Depth Measure for Dense Graphs Graphs Petr Hlinˇen´Y Faculty of Informatics, Masaryk University Brno, Czech Republic
    page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Shrub-Depth a successful depth measure for dense graphs graphs Petr Hlinˇen´y Faculty of Informatics, Masaryk University Brno, Czech Republic Ingredients: joint results with J. Gajarsk´y,R. Ganian, O. Kwon, J. Neˇsetˇril,J. Obdrˇz´alek, S. Ordyniak, P. Ossona de Mendez Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 1 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 Measuring Width or Depth? • Being close to a TREE { \•-width" sparse dense tree-width / branch-width { showing a structure clique-width / rank-width { showing a construction • Being close to a STAR { \•-depth" sparse dense tree-depth { containment in a structure ??? (will show) Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 2 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion". Petr Hlinˇen´y, Sparsity, Logic . , Warwick, 2018 3 / 19 Shrub-depth measure for dense graphs page.19 1 Recall: Width Measures Tree-width tw(G) ≤ k if whole G can be covered by bags of size ≤ k + 1, arranged in a \tree-like fashion".
    [Show full text]
  • A Fast Algorithm for the Maximum Clique Problem Patric R
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 120 (2002) 197–207 A fast algorithm for the maximum clique problem Patric R. J. Osterg%# ard ∗ Department of Computer Science and Engineering, Helsinki University of Technology, P.O. Box 5400, 02015 HUT, Finland Received 12 October 1999; received in revised form 29 May 2000; accepted 19 June 2001 Abstract Given a graph, in the maximum clique problem, one desires to ÿnd the largest number of vertices, any two of which are adjacent. A branch-and-bound algorithm for the maximum clique problem—which is computationally equivalent to the maximum independent (stable) set problem—is presented with the vertex order taken from a coloring of the vertices and with a new pruning strategy. The algorithm performs successfully for many instances when applied to random graphs and DIMACS benchmark graphs. ? 2002 Elsevier Science B.V. All rights reserved. 1. Introduction We denote an undirected graph by G =(V; E), where V is the set of vertices and E is the set of edges. Two vertices are said to be adjacent if they are connected by an edge. A clique of a graph is a set of vertices, any two of which are adjacent. Cliques with the following two properties have been studied over the last three decades: maximal cliques, whose vertices are not a subset of the vertices of a larger clique, and maximum cliques, which are the largest among all cliques in a graph (maximum cliques are clearly maximal).
    [Show full text]
  • Cliques and Clubs⋆
    Cliques and Clubs? Petr A. Golovach1, Pinar Heggernes1, Dieter Kratsch2, and Arash Rafiey1 1 Department of Informatics, University of Bergen, Norway, fpetr.golovach,pinar.heggernes,[email protected] 2 LITA, Universit´ede Lorraine - Metz, France, [email protected] Abstract. Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracted significant attention recently. We present new positive results for the problem on large and important graph classes. In particular we show that for input G and s, a maximum s-club in G can be computed in polynomial time when G is a chordal bipartite or a strongly chordal or a distance hereditary graph. On a superclass of these graphs, weakly chordal graphs, we obtain a polynomial-time algorithm when s is an odd integer, which is best possible as the problem is NP-hard on this clas for even values of s. We complement these results by proving the NP-hardness of the problem for every fixed s on 4-chordal graphs, a superclass of weakly chordal graphs. Finally, if G is an AT-free graph, we prove that the problem can be solved in polynomial time when s ≥ 2, which gives an interesting contrast to the fact that the problem is NP-hard for s = 1 on this graph class.
    [Show full text]
  • 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.
    [Show full text]
  • Clique Colourings of Geometric Graphs
    CLIQUE COLOURINGS OF GEOMETRIC GRAPHS COLIN MCDIARMID, DIETER MITSCHE, AND PAWELPRA LAT Abstract. A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x1;:::; xn in the plane, and a threshold r > 0, the corre- sponding geometric graph has vertex set fv1; : : : ; vng, and distinct vi and vj are adjacent when the Euclidean distance between xi and xj is at most r. We investigate the clique chromatic number of such graphs. We first show that the clique chromatic number is at most 9 for any geometric graph in the plane, and briefly consider geometric graphs in higher dimensions. Then we study the asymptotic behaviour of the clique chromatic number for the random geometric graph G(n; r) in the plane, where n random points are independently and uniformly distributed in a suitable square. We see that as r increases from 0, with high probability the clique chromatic number is 1 for very small r, then 2 for small r, then at least 3 for larger r, and finally drops back to 2. 1. Introduction and main results In this section we introduce clique colourings and geometric graphs; and we present our main results, on clique colourings of deterministic and random geometric graphs. Recall that a proper colouring of a graph is a labeling of its vertices with colours such that no two vertices sharing the same edge have the same colour; and the smallest number of colours in a proper colouring of a graph G = (V; E) is its chromatic number, denoted by χ(G).
    [Show full text]
  • Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses
    Algorithmic Graph Theory Part III Perfect Graphs and Their Subclasses Martin Milanicˇ [email protected] University of Primorska, Koper, Slovenia Dipartimento di Informatica Universita` degli Studi di Verona, March 2013 1/55 What we’ll do 1 THE BASICS. 2 PERFECT GRAPHS. 3 COGRAPHS. 4 CHORDAL GRAPHS. 5 SPLIT GRAPHS. 6 THRESHOLD GRAPHS. 7 INTERVAL GRAPHS. 2/55 THE BASICS. 2/55 Induced Subgraphs Recall: Definition Given two graphs G = (V , E) and G′ = (V ′, E ′), we say that G is an induced subgraph of G′ if V ⊆ V ′ and E = {uv ∈ E ′ : u, v ∈ V }. Equivalently: G can be obtained from G′ by deleting vertices. Notation: G < G′ 3/55 Hereditary Graph Properties Hereditary graph property (hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Formally: a set of graphs X such that G ∈ X and H < G ⇒ H ∈ X . 4/55 Hereditary Graph Properties Hereditary graph property (Hereditary graph class) = a class of graphs closed under deletion of vertices = a class of graphs closed under taking induced subgraphs Examples: forests complete graphs line graphs bipartite graphs planar graphs graphs of degree at most ∆ triangle-free graphs perfect graphs 5/55 Hereditary Graph Properties Why hereditary graph classes? Vertex deletions are very useful for developing algorithms for various graph optimization problems. Every hereditary graph property can be described in terms of forbidden induced subgraphs. 6/55 Hereditary Graph Properties H-free graph = a graph that does not contain H as an induced subgraph Free(H) = the class of H-free graphs Free(M) := H∈M Free(H) M-free graphT = a graph in Free(M) Proposition X hereditary ⇐⇒ X = Free(M) for some M M = {all (minimal) graphs not in X} The set M is the set of forbidden induced subgraphs for X.
    [Show full text]
  • Graph Theory
    Stephen P. Borgatti Graph Theory This is FIRST draft. Very likely to contain errors. lthough graph theory is one of the younger branches of mathematics, it is fundamental to a number of applied fields, including operations research, A computer science, and social network analysis. In this chapter we discuss the basic concepts of graph theory from the point of view of social network analysis. Graphs The fundamental concept of graph theory is the graph, which (despite the name) is best thought of as a mathematical object rather than a diagram, even though graphs have a very natural graphical representation. A graph – usually denoted G(V,E) or G = (V,E) – consists of set of vertices V together with a set of edges E. Vertices are also known as nodes, points and (in social networks) as actors, agents or players. Edges are also known as lines and (in social networks) as ties or links. An edge e = (u,v) is defined by the unordered pair of vertices that serve as its end points. Two vertices u and v are adjacent if there exists an edge (u,v) that connects them. An edge e = (u,u) that links a vertex to itself is known as a self-loop or reflexive tie. The number of vertices in a graph is usually denoted n while the number of edges is usually denoted m. As an example, the graph depicted in Figure 1 has vertex set V={a,b,c,d,e.f} and edge set E = {(a,b),(b,c),(c,d),(c,e),(d,e),(e,f)}.
    [Show full text]
  • Linear Clique-Width for Hereditary Classes of Cographs 2
    LINEAR CLIQUE-WIDTH FOR HEREDITARY CLASSES OF COGRAPHS Robert Brignall∗ Nicholas Korpelainen∗ Department of Mathematics and Statistics Mathematics Department The Open University University of Derby Milton Keynes, UK Derby, UK Vincent Vatter∗† Department of Mathematics University of Florida Gainesville, Florida USA The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique- width if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permu- tation classes. 1. INTRODUCTION A variety of measures of the complexity of graph classes have been introduced and have proved useful for algorithmic problems [3–5, 7, 8, 11, 17, 18, 22, 23]. We are concerned with clique-width, introduced by Courcelle, Engelfriet, and Rozenberg [6], and more pertinently, the linear version of this parameter, due to Lozin and Rautenbach [20], and studied further by Gurski and Wanke [14]. The linear clique-width of a graph G, lcw(G), is the size of the smallest alphabet Σ such that G can arXiv:1305.0636v5 [math.CO] 25 Jan 2016 be constructed by a sequence of the following three operations: • add a new vertex labeled by a letter in Σ, • add edges between all vertices labeled i and all vertices labeled j (for i 6= j), and • give all vertices labeled i the label j. ∗All three authors were partially supported by EPSRC via the grant EP/J006130/1. †Vatter’s research was also partially supported by the National Security Agency under Grant Number H98230-12-1-0207 and the National Science Foundation under Grant Number DMS-1301692.
    [Show full text]