Graph Classes and Forbidden Patterns on Three Vertices
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Forbidden Subgraph Characterization of Quasi-Line Graphs Medha Dhurandhar [email protected]
Forbidden Subgraph Characterization of Quasi-line Graphs Medha Dhurandhar [email protected] Abstract: Here in particular, we give a characterization of Quasi-line Graphs in terms of forbidden induced subgraphs. In general, we prove a necessary and sufficient condition for a graph to be a union of two cliques. 1. Introduction: A graph is a quasi-line graph if for every vertex v, the set of neighbours of v is expressible as the union of two cliques. Such graphs are more general than line graphs, but less general than claw-free graphs. In [2] Chudnovsky and Seymour gave a constructive characterization of quasi-line graphs. An alternative characterization of quasi-line graphs is given in [3] stating that a graph has a fuzzy reconstruction iff it is a quasi-line graph and also in [4] using the concept of sums of Hoffman graphs. Here we characterize quasi-line graphs in terms of the forbidden induced subgraphs like line graphs. We consider in this paper only finite, simple, connected, undirected graphs. The vertex set of G is denoted by V(G), the edge set by E(G), the maximum degree of vertices in G by Δ(G), the maximum clique size by (G) and the chromatic number by G). N(u) denotes the neighbourhood of u and N(u) = N(u) + u. For further notation please refer to Harary [3]. 2. Main Result: Before proving the main result we prove some lemmas, which will be used later. Lemma 1: If G is {3K1, C5}-free, then either 1) G ~ K|V(G)| or 2) If v, w V(G) are s.t. -
The AVD-Edge-Coloring Conjecture for Some Split Graphs
Matem´atica Contempor^anea, Vol. 44,1{10 c 2015, Sociedade Brasileira de Matem´atica The AVD-edge-coloring conjecture for some split graphs Alo´ısiode Menezes Vilas-B^oas C´eliaPicinin de Mello Abstract Let G be a simple graph. An adjacent vertex distinguishing edge- coloring (AVD-edge-coloring) of G is an edge-coloring of G such that for each pair of adjacent vertices u; v of G, the set of colors assigned to the edges incident with u differs from the set of colors assigned to the edges incident with v. The adjacent vertex distinguishing 0 chromatic index of G, denoted χa(G), is the minimum number of colors required to produce an AVD-edge-coloring for G. The AVD- edge-coloring conjecture states that every simple connected graph G ∼ 0 with at least three vertices and G =6 C5 has χa(G) ≤ ∆(G) + 2. The conjecture is open for arbitrary graphs, but it holds for some classes of graphs. In this note we focus on split graphs. We prove this AVD-edge- coloring conjecture for split-complete graphs and split-indifference graphs. 1 Introduction In this paper, G denotes a simple, undirected, finite, connected graph. The sets V (G) and E(G) are the vertex and edge sets of G. Let u; v 2 2000 AMS Subject Classification: 05C15. Key Words and Phrases: edge-coloring, adjacent strong edge-coloring, split graph. Supported by CNPq (132194/2010-4 and 308314/2013-1). The AVD-edge-coloring conjecture for some split graphs 2 V (G). We denote an edge by uv.A clique is a set of vertices pairwise adjacent in G and a stable set is a set of vertices such that no two of which are adjacent. -
Infinitely Many Minimal Classes of Graphs of Unbounded Clique-Width∗
Infinitely many minimal classes of graphs of unbounded clique-width∗ A. Collins, J. Foniok†, N. Korpelainen, V. Lozin, V. Zamaraev Abstract The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width. Keywords: clique-width, linear clique-width, hereditary class 1 Introduction Clique-width is a graph parameter which is important in theoretical computer science, because many algorithmic problems that are generally NP-hard become polynomial-time solvable when restricted to graphs of bounded clique-width [4]. -
Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated. -
Counting Independent Sets in Graphs with Bounded Bipartite Pathwidth∗
Counting independent sets in graphs with bounded bipartite pathwidth∗ Martin Dyery Catherine Greenhillz School of Computing School of Mathematics and Statistics University of Leeds UNSW Sydney, NSW 2052 Leeds LS2 9JT, UK Australia [email protected] [email protected] Haiko M¨uller∗ School of Computing University of Leeds Leeds LS2 9JT, UK [email protected] 7 August 2019 Abstract We show that a simple Markov chain, the Glauber dynamics, can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair's work on approximately counting matchings, that is, independent sets in line graphs. The class of graphs with bounded bipartite pathwidth includes claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth. We also show how to extend all our results to polynomially-bounded vertex weights. 1 Introduction There is a well-known bijection between matchings of a graph G and independent sets in the line graph of G. We will show that we can approximate the number of independent sets ∗A preliminary version of this paper appeared as [19]. yResearch supported by EPSRC grant EP/S016562/1 \Sampling in hereditary classes". zResearch supported by Australian Research Council grant DP190100977. 1 in graphs for which all bipartite induced subgraphs are well structured, in a sense that we will define precisely. -
Lecture 10: April 20, 2005 Perfect Graphs
Re-revised notes 4-22-2005 10pm CMSC 27400-1/37200-1 Combinatorics and Probability Spring 2005 Lecture 10: April 20, 2005 Instructor: L´aszl´oBabai Scribe: Raghav Kulkarni TA SCHEDULE: TA sessions are held in Ryerson-255, Monday, Tuesday and Thursday 5:30{6:30pm. INSTRUCTOR'S EMAIL: [email protected] TA's EMAIL: [email protected], [email protected] IMPORTANT: Take-home test Friday, April 29, due Monday, May 2, before class. Perfect Graphs k 1=k Shannon capacity of a graph G is: Θ(G) := limk (α(G )) : !1 Exercise 10.1 Show that α(G) χ(G): (G is the complement of G:) ≤ Exercise 10.2 Show that χ(G H) χ(G)χ(H): · ≤ Exercise 10.3 Show that Θ(G) χ(G): ≤ So, α(G) Θ(G) χ(G): ≤ ≤ Definition: G is perfect if for all induced sugraphs H of G, α(H) = χ(H); i. e., the chromatic number is equal to the clique number. Theorem 10.4 (Lov´asz) G is perfect iff G is perfect. (This was open under the name \weak perfect graph conjecture.") Corollary 10.5 If G is perfect then Θ(G) = α(G) = χ(G): Exercise 10.6 (a) Kn is perfect. (b) All bipartite graphs are perfect. Exercise 10.7 Prove: If G is bipartite then G is perfect. Do not use Lov´asz'sTheorem (Theorem 10.4). 1 Lecture 10: April 20, 2005 2 The smallest imperfect (not perfect) graph is C5 : α(C5) = 2; χ(C5) = 3: For k 2, C2k+1 imperfect. -
I?'!'-Comparability Graphs
Discrete Mathematics 74 (1989) 173-200 173 North-Holland I?‘!‘-COMPARABILITYGRAPHS C.T. HOANG Department of Computer Science, Rutgers University, New Brunswick, NJ 08903, U.S.A. and B.A. REED Discrete Mathematics Group, Bell Communications Research, Morristown, NJ 07950, U.S.A. In 1981, Chv&tal defined the class of perfectly orderable graphs. This class of perfect graphs contains the comparability graphs. In this paper, we introduce a new class of perfectly orderable graphs, the &comparability graphs. This class generalizes comparability graphs in a natural way. We also prove a decomposition theorem which leads to a structural characteriza- tion of &comparability graphs. Using this characterization, we develop a polynomial-time recognition algorithm and polynomial-time algorithms for the clique and colouring problems for &comparability graphs. 1. Introduction A natural way to color the vertices of a graph is to put them in some order v*cv~<” l < v, and then assign colors in the following manner: (1) Consider the vertxes sequentially (in the sequence given by the order) (2) Assign to each Vi the smallest color (positive integer) not used on any neighbor vi of Vi with j c i. We shall call this the greedy coloring algorithm. An order < of a graph G is perfect if for each subgraph H of G, the greedy algorithm applied to H, gives an optimal coloring of H. An obstruction in a ordered graph (G, <) is a set of four vertices {a, b, c, d} with edges ab, bc, cd (and no other edge) and a c b, d CC. Clearly, if < is a perfect order on G, then (G, C) contains no obstruction (because the chromatic number of an obstruction is twu, but the greedy coloring algorithm uses three colors). -
Proper Interval Graphs and the Guard Problem 1
DISCRETE N~THEMATICS ELSEVIER Discrete Mathematics 170 (1997) 223-230 Note Proper interval graphs and the guard problem1 Chiuyuan Chen*, Chin-Chen Chang, Gerard J. Chang Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan Received 26 September 1995; revised 27 August 1996 Abstract This paper is a study of the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. We prove that proper interval graphs with ~> 2 vertices have hamiltonian paths, those with ~>3 vertices have hamiltonian cycles, and those with />4 vertices are hamiltonian-connected if and only if they are, respectively, 1-, 2-, or 3-connected. We also study the guard problem in spiral polygons by connecting the class of nontrivial connected proper interval graphs with the class of stick-intersection graphs of spiral polygons. Keywords." Proper interval graph; Hamiltonian path (cycle); Hamiltonian-connected; Guard; Visibility; Spiral polygon I. Introduction The main purpose of this paper is to study the hamiltonicity of proper interval graphs with applications to the guard problem in spiral polygons. Our terminology and graph notation are standard, see [2], except as indicated. The intersection graph of a family ~ of nonempty sets is derived by representing each set in ~ with a vertex and connecting two vertices with an edge if and only if their corresponding sets intersect. An interval graph is the intersection graph G of a family J of intervals on a real line. J is usually called the interval model for G. A proper interval graph is an interval graph with an interval model ~¢ such that no interval in J properly contains another. -
Density Theorems for Bipartite Graphs and Related Ramsey-Type Results
Density theorems for bipartite graphs and related Ramsey-type results Jacob Fox Benny Sudakov Princeton UCLA and IAS Ramsey’s theorem Definition: r(G) is the minimum N such that every 2-edge-coloring of the complete graph KN contains a monochromatic copy of graph G. Theorem: (Ramsey-Erdos-Szekeres,˝ Erdos)˝ t/2 2t 2 ≤ r(Kt ) ≤ 2 . Question: (Burr-Erd˝os1975) How large is r(G) for a sparse graph G on n vertices? Ramsey numbers for sparse graphs Conjecture: (Burr-Erd˝os1975) For every d there exists a constant cd such that if a graph G has n vertices and maximum degree d, then r(G) ≤ cd n. Theorem: 1 (Chv´atal-R¨odl-Szemer´edi-Trotter 1983) cd exists. 2αd 2 (Eaton 1998) cd ≤ 2 . βd αd log2 d 3 (Graham-R¨odl-Ruci´nski2000) 2 ≤ cd ≤ 2 . Moreover, if G is bipartite, r(G) ≤ 2αd log d n. Density theorem for bipartite graphs Theorem: (F.-Sudakov) Let G be a bipartite graph with n vertices and maximum degree d 2 and let H be a bipartite graph with parts |V1| = |V2| = N and εN edges. If N ≥ 8dε−d n, then H contains G. Corollary: For every bipartite graph G with n vertices and maximum degree d, r(G) ≤ d2d+4n. (D. Conlon independently proved that r(G) ≤ 2(2+o(1))d n.) Proof: Take ε = 1/2 and H to be the graph of the majority color. Ramsey numbers for cubes Definition: d The binary cube Qd has vertex set {0, 1} and x, y are adjacent if x and y differ in exactly one coordinate. -
Classes of Graphs with Restricted Interval Models Andrzej Proskurowski, Jan Arne Telle
Classes of graphs with restricted interval models Andrzej Proskurowski, Jan Arne Telle To cite this version: Andrzej Proskurowski, Jan Arne Telle. Classes of graphs with restricted interval models. Discrete Mathematics and Theoretical Computer Science, DMTCS, 1999, 3 (4), pp.167-176. hal-00958935 HAL Id: hal-00958935 https://hal.inria.fr/hal-00958935 Submitted on 13 Mar 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. Discrete Mathematics and Theoretical Computer Science 3, 1999, 167–176 Classes of graphs with restricted interval models ✁ Andrzej Proskurowski and Jan Arne Telle ✂ University of Oregon, Eugene, Oregon, USA ✄ University of Bergen, Bergen, Norway received June 30, 1997, revised Feb 18, 1999, accepted Apr 20, 1999. We introduce ☎ -proper interval graphs as interval graphs with interval models in which no interval is properly con- tained in more than ☎ other intervals, and also provide a forbidden induced subgraph characterization of this class of ✆✞✝✠✟ graphs. We initiate a graph-theoretic study of subgraphs of ☎ -proper interval graphs with maximum clique size and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter ✆ ✆ ☎ to vary from 0 to , we obtain a nested hierarchy of graph families, from graphs of bandwidth at most to graphs of pathwidth at most ✆ . -
Matroid Theory
MATROID THEORY HAYLEY HILLMAN 1 2 HAYLEY HILLMAN Contents 1. Introduction to Matroids 3 1.1. Basic Graph Theory 3 1.2. Basic Linear Algebra 4 2. Bases 5 2.1. An Example in Linear Algebra 6 2.2. An Example in Graph Theory 6 3. Rank Function 8 3.1. The Rank Function in Graph Theory 9 3.2. The Rank Function in Linear Algebra 11 4. Independent Sets 14 4.1. Independent Sets in Graph Theory 14 4.2. Independent Sets in Linear Algebra 17 5. Cycles 21 5.1. Cycles in Graph Theory 22 5.2. Cycles in Linear Algebra 24 6. Vertex-Edge Incidence Matrix 25 References 27 MATROID THEORY 3 1. Introduction to Matroids A matroid is a structure that generalizes the properties of indepen- dence. Relevant applications are found in graph theory and linear algebra. There are several ways to define a matroid, each relate to the concept of independence. This paper will focus on the the definitions of a matroid in terms of bases, the rank function, independent sets and cycles. Throughout this paper, we observe how both graphs and matrices can be viewed as matroids. Then we translate graph theory to linear algebra, and vice versa, using the language of matroids to facilitate our discussion. Many proofs for the properties of each definition of a matroid have been omitted from this paper, but you may find complete proofs in Oxley[2], Whitney[3], and Wilson[4]. The four definitions of a matroid introduced in this paper are equiv- alent to each other. -
A Class of Hypergraphs That Generalizes Chordal Graphs
MATH. SCAND. 106 (2010), 50–66 A CLASS OF HYPERGRAPHS THAT GENERALIZES CHORDAL GRAPHS ERIC EMTANDER Abstract In this paper we introduce a class of hypergraphs that we call chordal. We also extend the definition of triangulated hypergraphs, given by H. T. Hà and A. Van Tuyl, so that a triangulated hypergraph, according to our definition, is a natural generalization of a chordal (rigid circuit) graph. R. Fröberg has showed that the chordal graphs corresponds to graph algebras, R/I(G), with linear resolu- tions. We extend Fröberg’s method and show that the hypergraph algebras of generalized chordal hypergraphs, a class of hypergraphs that includes the chordal hypergraphs, have linear resolutions. The definitions we give, yield a natural higher dimensional version of the well known flag property of simplicial complexes. We obtain what we call d-flag complexes. 1. Introduction and preliminaries Let X be a finite set and E ={E1,...,Es } a finite collection of non empty subsets of X . The pair H = (X, E ) is called a hypergraph. The elements of X and E , respectively, are called the vertices and the edges, respectively, of the hypergraph. If we want to specify what hypergraph we consider, we may write X (H ) and E (H ) for the vertices and edges respectively. A hypergraph is called simple if: (1) |Ei |≥2 for all i = 1,...,s and (2) Ej ⊆ Ei only if i = j. If the cardinality of X is n we often just use the set [n] ={1, 2,...,n} instead of X . Let H be a hypergraph. A subhypergraph K of H is a hypergraph such that X (K ) ⊆ X (H ), and E (K ) ⊆ E (H ).IfY ⊆ X , the induced hypergraph on Y, HY , is the subhypergraph with X (HY ) = Y and with E (HY ) consisting of the edges of H that lie entirely in Y.