Graph Theory
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Directed Graphs Definition: an Directed Graph (Or Digraph) G = (V, E) Consists of a Nonempty Set V of Vertices (Or Nodes) and a Set E of Directed Edges (Or Arcs)
Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems (not currently included in overheads) Planar Graphs (not currently included in overheads) Graph Coloring (not currently included in overheads) Section 10.1 Section Summary Introduction to Graphs Graph Taxonomy Graph Models Graphs Definition: A graph G = (V, E) consists of a nonempty set V of vertices (or nodes) and a set E of edges. Each edge has either one or two vertices associated with it, called its endpoints. An edge is said to connect its endpoints. Example: a b This is a graph with four vertices and five edges. d c Remarks: The graphs we study here are unrelated to graphs of functions studied in Chapter 2. We have a lot of freedom when we draw a picture of a graph. All that matters is the connections made by the edges, not the particular geometry depicted. For example, the lengths of edges, whether edges cross, how vertices are depicted, and so on, do not matter A graph with an infinite vertex set is called an infinite graph. A graph with a finite vertex set is called a finite graph. We (following the text) restrict our attention to finite graphs. Some Terminology In a simple graph each edge connects two different vertices and no two edges connect the same pair of vertices. Multigraphs may have multiple edges connecting the same two vertices. When m different edges connect the vertices u and v, we say that {u,v} is an edge of multiplicity m. -
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. -
K-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees
k-Outerplanar Graphs, Planar Duality, and Low Stretch Spanning Trees Yuval Emek∗ Abstract Low distortion probabilistic embedding of graphs into approximating trees is an extensively studied topic. Of particular interest is the case where the approximating trees are required to be (subgraph) spanning trees of the given graph (or multigraph), in which case, the focus is usually on the equivalent problem of finding a (single) tree with low average stretch. Among the classes of graphs that received special attention in this context are k-outerplanar graphs (for a fixed k): Chekuri, Gupta, Newman, Rabinovich, and Sinclair show that every k-outerplanar graph can be probabilistically embedded into approximating trees with constant distortion regardless of the size of the graph. The approximating trees in the technique of Chekuri et al. are not necessarily spanning trees, though. In this paper it is shown that every k-outerplanar multigraph admits a spanning tree with constant average stretch. This immediately translates to a constant bound on the distortion of probabilistically embedding k-outerplanar graphs into their spanning trees. Moreover, an efficient randomized algorithm is presented for constructing such a low average stretch spanning tree. This algorithm relies on some new insights regarding the connection between low average stretch spanning trees and planar duality. Keywords: planar graphs, outerplanarity, average stretch, planar dual. ∗Microsoft Israel R&D Center, Herzelia, Israel and School of Electrical Engineering, Tel Aviv University, Tel Aviv, Israel. E-mail: [email protected]. Supported in part by the Israel Science Foundation, grants 221/07 and 664/05. 1 Introduction The problem. -
Planar Graph Theory We Say That a Graph Is Planar If It Can Be Drawn in the Plane Without Edges Crossing
Planar Graph Theory We say that a graph is planar if it can be drawn in the plane without edges crossing. We use the term plane graph to refer to a planar depiction of a planar graph. e.g. K4 is a planar graph Q1: The following is also planar. Find a plane graph version of the graph. A B F E D C A Method that sometimes works for drawing the plane graph for a planar graph: 1. Find the largest cycle in the graph. 2. The remaining edges must be drawn inside/outside the cycle so that they do not cross each other. Q2: Using the method above, find a plane graph version of the graph below. A B C D E F G H non e.g. K3,3: K5 Here are three (plane graph) depictions of the same planar graph: J N M J K J N I M K K I N M I O O L O L L A face of a plane graph is a region enclosed by the edges of the graph. There is also an unbounded face, which is the outside of the graph. Q3: For each of the plane graphs we have drawn, find: V = # of vertices of the graph E = # of edges of the graph F = # of faces of the graph Q4: Do you have a conjecture for an equation relating V, E and F for any plane graph G? Q5: Can you name the 5 Platonic Solids (i.e. regular polyhedra)? (This is a geometry question.) Q6: Find the # of vertices, # of edges and # of faces for each Platonic Solid. -
Adjacency and Incidence Matrices
Adjacency and Incidence Matrices 1 / 10 The Incidence Matrix of a Graph Definition Let G = (V ; E) be a graph where V = f1; 2;:::; ng and E = fe1; e2;:::; emg. The incidence matrix of G is an n × m matrix B = (bik ), where each row corresponds to a vertex and each column corresponds to an edge such that if ek is an edge between i and j, then all elements of column k are 0 except bik = bjk = 1. 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 2 / 10 The First Theorem of Graph Theory Theorem If G is a multigraph with no loops and m edges, the sum of the degrees of all the vertices of G is 2m. Corollary The number of odd vertices in a loopless multigraph is even. 3 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. Then the rank of B is n − 1 if G is bipartite and n otherwise. Example 1 2 e 21 1 13 f 61 0 07 3 B = 6 7 g 40 1 05 4 0 0 1 4 / 10 Linear Algebra and Incidence Matrices of Graphs Recall that the rank of a matrix is the dimension of its row space. Proposition Let G be a connected graph with n vertices and let B be the incidence matrix of G. -
Math 7410 Graph Theory
Math 7410 Graph Theory Bogdan Oporowski Department of Mathematics Louisiana State University April 14, 2021 Definition of a graph Definition 1.1 A graph G is a triple (V,E, I) where ◮ V (or V (G)) is a finite set whose elements are called vertices; ◮ E (or E(G)) is a finite set disjoint from V whose elements are called edges; and ◮ I, called the incidence relation, is a subset of V E in which each edge is × in relation with exactly one or two vertices. v2 e1 v1 Example 1.2 ◮ V = v1, v2, v3, v4 e5 { } e2 e4 ◮ E = e1,e2,e3,e4,e5,e6,e7 { } e6 ◮ I = (v1,e1), (v1,e4), (v1,e5), (v1,e6), { v4 (v2,e1), (v2,e2), (v3,e2), (v3,e3), (v3,e5), e7 v3 e3 (v3,e6), (v4,e3), (v4,e4), (v4,e7) } Simple graphs Definition 1.3 ◮ Edges incident with just one vertex are loops. ◮ Edges incident with the same pair of vertices are parallel. ◮ Graphs with no parallel edges and no loops are called simple. v2 e1 v1 e5 e2 e4 e6 v4 e7 v3 e3 Edges of a simple graph can be described as v e1 v 2 1 two-element subsets of the vertex set. Example 1.4 e5 e2 e4 E = v1, v2 , v2, v3 , v3, v4 , e6 {{ } { } { } v1, v4 , v1, v3 . v4 { } { }} v e7 3 e3 Note 1.5 Graph Terminology Definition 1.6 ◮ The graph G is empty if V = , and is trivial if E = . ∅ ∅ ◮ The cardinality of the vertex-set of a graph G is called the order of G and denoted G . -
Network Analysis of the Multimodal Freight Transportation System in New York City
Network Analysis of the Multimodal Freight Transportation System in New York City Project Number: 15 – 2.1b Year: 2015 FINAL REPORT June 2018 Principal Investigator Qian Wang Researcher Shuai Tang MetroFreight Center of Excellence University at Buffalo Buffalo, NY 14260-4300 Network Analysis of the Multimodal Freight Transportation System in New York City ABSTRACT The research is aimed at examining the multimodal freight transportation network in the New York metropolitan region to identify critical links, nodes and terminals that affect last-mile deliveries. Two types of analysis were conducted to gain a big picture of the region’s freight transportation network. First, three categories of network measures were generated for the highway network that carries the majority of last-mile deliveries. They are the descriptive measures that demonstrate the basic characteristics of the highway network, the network structure measures that quantify the connectivity of nodes and links, and the accessibility indices that measure the ease to access freight demand, services and activities. Second, 71 multimodal freight terminals were selected and evaluated in terms of their accessibility to major multimodal freight demand generators such as warehousing establishments. As found, the most important highways nodes that are critical in terms of connectivity and accessibility are those in and around Manhattan, particularly the bridges and tunnels connecting Manhattan to neighboring areas. Major multimodal freight demand generators, such as warehousing establishments, have better accessibility to railroad and marine port terminals than air and truck terminals in general. The network measures and findings in the research can be used to understand the inventory of the freight network in the system and to conduct freight travel demand forecasting analysis. -
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. -
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 ∨ . -
Matroids You Have Known
26 MATHEMATICS MAGAZINE Matroids You Have Known DAVID L. NEEL Seattle University Seattle, Washington 98122 [email protected] NANCY ANN NEUDAUER Pacific University Forest Grove, Oregon 97116 nancy@pacificu.edu Anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day. —Gian Carlo Rota [10] Why matroids? Have you noticed hidden connections between seemingly unrelated mathematical ideas? Strange that finding roots of polynomials can tell us important things about how to solve certain ordinary differential equations, or that computing a determinant would have anything to do with finding solutions to a linear system of equations. But this is one of the charming features of mathematics—that disparate objects share similar traits. Properties like independence appear in many contexts. Do you find independence everywhere you look? In 1933, three Harvard Junior Fellows unified this recurring theme in mathematics by defining a new mathematical object that they dubbed matroid [4]. Matroids are everywhere, if only we knew how to look. What led those junior-fellows to matroids? The same thing that will lead us: Ma- troids arise from shared behaviors of vector spaces and graphs. We explore this natural motivation for the matroid through two examples and consider how properties of in- dependence surface. We first consider the two matroids arising from these examples, and later introduce three more that are probably less familiar. Delving deeper, we can find matroids in arrangements of hyperplanes, configurations of points, and geometric lattices, if your tastes run in that direction. -
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges
Characterizations of Restricted Pairs of Planar Graphs Allowing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen Kobourov1, and Michael Schulz2 1 University of Arizona, USA {jfowler,kobourov}@cs.arizona.edu ⋆ 2 University of Cologne, Germany {mjuenger,schulz}@informatik.uni-koeln.de ⋆⋆ Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient al- gorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale integration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can be used with multiple edge sets that each correspond to different edge colors or circuit layers. While the pairwise union of all edge sets may be nonplanar, a planar drawing of each layer may be possible, as crossings between edges of distinct edge sets are permitted.