CLRS B.4 Graph Theory Definitions Unit 1: DFS Informally, a Graph
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Strongly Connected Component
Graph IV Ian Things that we would talk about ● DFS ● Tree ● Connectivity Useful website http://codeforces.com/blog/entry/16221 Recommended Practice Sites ● HKOJ ● Codeforces ● Topcoder ● Csacademy ● Atcoder ● USACO ● COCI Term in Directed Tree ● Consider node 4 – Node 2 is its parent – Node 1, 2 is its ancestors – Node 5 is its sibling – Node 6 is its child – Node 6, 7, 8 is its descendants ● Node 1 is the root DFS Forest ● When we do DFS on a graph, we would obtain a DFS forest. Noted that the graph is not necessarily a tree. ● Some of the information we get through the DFS is actually very useful, such as – Starting time of a node – Finishing time of a node – Parent of the node Some Tricks Using DFS Order ● Suppose vertex v is ancestor(not only parent) of u – Starting time of v < starting time of u – Finishing time of v > starting time of u ● st[v] < st[u] <= ft[u] < ft[v] ● O(1) to check if ancestor or not ● Flatten the tree to store subtree information(maybe using segment tree or other data structure to maintain) ● Super useful !!!!!!!!!! Partial Sum on Tree ● Given queries, each time increase all node from node v to node u by 1 ● Assume node v is ancestor of node u ● sum[u]++, sum[par[v]]-- ● Run dfs in root dfs(v) for all child u dfs(u) d[v] = d[v] + d[u] Types of Edges ● Tree edges – Edges that forms a tree ● Forward edges – Edges that go from a node to its descendants but itself is not a tree edge. -
1 Vertex Connectivity 2 Edge Connectivity 3 Biconnectivity
1 Vertex Connectivity So far we've talked about connectivity for undirected graphs and weak and strong connec- tivity for directed graphs. For undirected graphs, we're going to now somewhat generalize the concept of connectedness in terms of network robustness. Essentially, given a graph, we may want to answer the question of how many vertices or edges must be removed in order to disconnect the graph; i.e., break it up into multiple components. Formally, for a connected graph G, a set of vertices S ⊆ V (G) is a separating set if subgraph G − S has more than one component or is only a single vertex. The set S is also called a vertex separator or a vertex cut. The connectivity of G, κ(G), is the minimum size of any S ⊆ V (G) such that G − S is disconnected or has a single vertex; such an S would be called a minimum separator. We say that G is k-connected if κ(G) ≥ k. 2 Edge Connectivity We have similar concepts for edges. For a connected graph G, a set of edges F ⊆ E(G) is a disconnecting set if G − F has more than one component. If G − F has two components, F is also called an edge cut. The edge-connectivity if G, κ0(G), is the minimum size of any F ⊆ E(G) such that G − F is disconnected; such an F would be called a minimum cut.A bond is a minimal non-empty edge cut; note that a bond is not necessarily a minimum cut. -
Efficient Multicore Algorithms for Identifying Biconnected Components
International Journal of Networking and Computing { www.ijnc.org ISSN 2185-2839 (print) ISSN 2185-2847 (online) Volume 6, Number 1, pages 87{106, January 2016 Efficient Multicore Algorithms For Identifying Biconnected Components1 Meher Chaitanya [email protected] and Kishore Kothapalli [email protected] International Institute of Information Technology Hyderabad, Gachibowli, India 500 032. Received: July 30, 2015 Revised: October 26, 2015 Accepted: December 1, 2015 Communicated by Akihiro Fujiwara Abstract In this paper we design and implement an algorithm for finding the biconnected components of a given graph. Our algorithm is based on experimental evidence that finding the bridges of a graph is usually easier and faster in the parallel setting. We use this property to first decompose the graph into independent and maximal 2-edge-connected subgraphs. To identify the articulation points in these 2-edge connected subgraphs, we again convert this into a problem of finding the bridges on an auxiliary graph. It is interesting to note that during the conversion process, the size of the graph may increase. However, we show that this small increase in size and the run time is offset by the consideration that finding bridges is easier in a parallel setting. We implement our algorithm on an Intel i7 980X CPU running 12 threads. We show that our algorithm is on average 2.45x faster than the best known current algorithms implemented on the same platform. Finally, we extend our approach to dense graphs by applying the sparsification technique suggested by Cong and Bader in [7]. Keywords: Biconnected components, Least common ancestor, 2-edge connected components, Articulation points 1 Introduction The biconnected components of a given graph are its maximal 2-connected subgraphs. -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
Vizing's Theorem and Edge-Chromatic Graph
VIZING'S THEOREM AND EDGE-CHROMATIC GRAPH THEORY ROBERT GREEN Abstract. This paper is an expository piece on edge-chromatic graph theory. The central theorem in this subject is that of Vizing. We shall then explore the properties of graphs where Vizing's upper bound on the chromatic index is tight, and graphs where the lower bound is tight. Finally, we will look at a few generalizations of Vizing's Theorem, as well as some related conjectures. Contents 1. Introduction & Some Basic Definitions 1 2. Vizing's Theorem 2 3. General Properties of Class One and Class Two Graphs 3 4. The Petersen Graph and Other Snarks 4 5. Generalizations and Conjectures Regarding Vizing's Theorem 6 Acknowledgments 8 References 8 1. Introduction & Some Basic Definitions Definition 1.1. An edge colouring of a graph G = (V; E) is a map C : E ! S, where S is a set of colours, such that for all e; f 2 E, if e and f share a vertex, then C(e) 6= C(f). Definition 1.2. The chromatic index of a graph χ0(G) is the minimum number of colours needed for a proper colouring of G. Definition 1.3. The degree of a vertex v, denoted by d(v), is the number of edges of G which have v as a vertex. The maximum degree of a graph is denoted by ∆(G) and the minimum degree of a graph is denoted by δ(G). Vizing's Theorem is the central theorem of edge-chromatic graph theory, since it provides an upper and lower bound for the chromatic index χ0(G) of any graph G. -
Superposition and Constructions of Graphs Without Nowhere-Zero K-flows
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Europ. J. Combinatorics (2002) 23, 281–306 doi:10.1006/eujc.2001.0563 Available online at http://www.idealibrary.com on Superposition and Constructions of Graphs Without Nowhere-zero k-flows M ARTIN KOCHOL Using multi-terminal networks we build methods on constructing graphs without nowhere-zero group- and integer-valued flows. In this way we unify known constructions of snarks (nontrivial cubic graphs without edge-3-colorings, or equivalently, without nowhere-zero 4-flows) and provide new ones in the same process. Our methods also imply new complexity results about nowhere-zero flows in graphs and state equivalences of Tutte’s 3- and 5-flow conjectures with formally weaker statements. c 2002 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION Nowhere-zero flows in graphs have been introduced by Tutte [38–40]. Primarily he showed that a planar graph is face-k-colorable if and only if it admits a nowhere-zero k-flow (its edges can be oriented and assigned values ±1,..., ±(k − 1) so that the sum of the incoming values equals the sum of the outcoming ones for every vertex of the graph). Tutte also proved the classical equivalence result that a graph admits a nowhere-zero k-flow if and only if it admits a flow whose values are the nonzero elements of a finite abelian group of order k. Seymour [35] has proved that every bridgeless graph admits a nowhere-zero 6-flow, thereby improving the 8-flow theorem of Jaeger [16] and Kilpatrick [20]. -
Finding Articulation Points and Bridges Articulation Points Articulation Point
Finding Articulation Points and Bridges Articulation Points Articulation Point Articulation Point A vertex v is an articulation point (also called cut vertex) if removing v increases the number of connected components. A graph with two articulation points. 3 / 1 Articulation Points Given I An undirected, connected graph G = (V; E) I A DFS-tree T with the root r Lemma A DFS on an undirected graph does not produce any cross edges. Conclusion I If a descendant u of a vertex v is adjacent to a vertex w, then w is a descendant or ancestor of v. 4 / 1 Removing a Vertex v Assume, we remove a vertex v 6= r from the graph. Case 1: v is an articulation point. I There is a descendant u of v which is no longer reachable from r. I Thus, there is no edge from the tree containing u to the tree containing r. Case 2: v is not an articulation point. I All descendants of v are still reachable from r. I Thus, for each descendant u, there is an edge connecting the tree containing u with the tree containing r. 5 / 1 Removing a Vertex v Problem I v might have multiple subtrees, some adjacent to ancestors of v, and some not adjacent. Observation I A subtree is not split further (we only remove v). Theorem A vertex v is articulation point if and only if v has a child u such that neither u nor any of u's descendants are adjacent to an ancestor of v. Question I How do we determine this efficiently for all vertices? 6 / 1 Detecting Descendant-Ancestor Adjacency Lowpoint The lowpoint low(v) of a vertex v is the lowest depth of a vertex which is adjacent to v or a descendant of v. -
Strongly Connected Components and Biconnected Components
Strongly Connected Components and Biconnected Components Daniel Wisdom 27 January 2017 1 2 3 4 5 6 7 8 Figure 1: A directed graph. Credit: Crash Course Coding Companion. 1 Strongly Connected Components A strongly connected component (SCC) is a set of vertices where every vertex can reach every other vertex. In an undirected graph the SCCs are just the groups of connected vertices. We can find them using a simple DFS. This works because, in an undirected graph, u can reach v if and only if v can reach u. This is not true in a directed graph, which makes finding the SCCs harder. More on that later. 2 Kernel graph We can travel between any two nodes in the same strongly connected component. So what if we replaced each strongly connected component with a single node? This would give us what we call the kernel graph, which describes the edges between strongly connected components. Note that the kernel graph is a directed acyclic graph because any cycles would constitute an SCC, so the cycle would be compressed into a kernel. 3 1,2,5,6 4,7 8 Figure 2: Kernel graph of the above directed graph. 3 Tarjan's SCC Algorithm In a DFS traversal of the graph every SCC will be a subtree of the DFS tree. This subtree is not necessarily proper, meaning that it may be missing some branches which are their own SCCs. This means that if we can identify every node that is the root of its SCC in the DFS, we can enumerate all the SCCs. -
Characterizing Simultaneous Embedding with Fixed Edges
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by computer science publication server Characterizing Simultaneous Embedding with Fixed Edges J. Joseph Fowler1, Michael J¨unger2, Stephen G. 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 plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawings. While any number of planar graphs have a simultaneous embedding without fixed edges, determining which graphs always share a simultaneous embedding with fixed edges (SEFE) has been open. We partially close this problem by giving a necessary condition to determine when pairs of graphs have a SEFE. 1 Introduction In many practical applications including the visualization of large graphs and very-large-scale in- tegration (VLSI) of circuits on the same chip, edge crossings are undesirable. A single vertex set can suffice in which multiple edge sets correspond to different edge colors or circuit layers. While the union of any pair of edge sets may be nonplanar, a planar drawing of each layer may still be possible, as crossings between edges of distinct edge sets is permitted. This corresponds to the problem of simultaneous embedding (SE). Without restrictions on the types of edges used, this problem is trivial since any number of planar graphs can be drawn on the same fixed set of vertex locations [6]. -
Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs
Near-Linear Time Constant-Factor Approximation Algorithm for Branch-Decomposition of Planar Graphs1 Qian-Ping Gu and Gengchun Xu School of Computing Science, Simon Fraser University Burnaby BC Canada V5A1S6 [email protected],[email protected] Abstract: We give an algorithm which for an input planar graph G of n vertices and integer k, in min{O(n log3 n),O(nk2)} time either constructs a branch-decomposition of G k+1 with width at most (2 + δ)k, δ > 0 is a constant, or a (k + 1) × ⌈ 2 ⌉ cylinder minor of G implying bw(G) >k, bw(G) is the branchwidth of G. This is the first O˜(n) time constant- factor approximation for branchwidth/treewidth and largest grid/cylinder minors of planar graphs and improves the previous min{O(n1+ǫ),O(nk2)} (ǫ> 0 is a constant) time constant- factor approximations. For a planar graph G and k = bw(G), a branch-decomposition of g k width at most (2 + δ)k and a g × 2 cylinder/grid minor with g = β , β > 2 is constant, can be computed by our algorithm in min{O(n log3 n log k),O(nk2 log k)} time. Key words: Branch-/tree-decompositions, grid minor, planar graphs, approximation algo- rithm. 1 Introduction The notions of branchwidth and branch-decomposition introduced by Robertson and Sey- mour [31] in relation to the notions of treewidth and tree-decomposition have important algorithmic applications. The branchwidth bw(G) and the treewidth tw(G) of graph G 3 are linearly related: max{bw(G), 2} ≤ tw(G)+1 ≤ max{⌊ 2 bw(G)⌋, 2} for every G with more than one edge, and there are simple translations between branch-decompositions and tree-decompositions that meet the linear relations [31]. -
Number One Is in NC
数理解析研究所講究録 第 790 巻 1992 年 155-161 155 Deciding whether Graph $G$ Has Page Number One is in NC 増山 繁 (豊橋技術科学大学知識情報工学系) Shigeru MASUYAMA Department of Knowledge-Based Information Engineering Toyohashi University of Technology Toyohashi 441, Japan 内藤昭三 (NTT 基礎研究所) Shozo NAITO Basic Research Laboratories NTT Musashino 180, Japan Abstract Based on a forbidden subgraph characterization of a graph to have one page, we develop a polylog time algorithm to tell if page number of given graph $G$ is one with polynomial number of processors, clarifying this problem to be in NC. 1 Introduction is crossing when they are drawn on the same side of the linear arrangement of vertices. Similar problem setting ap- This paper discusses the problem of pears in the formulation of the non- deciding whether the given graph is crossing constraint on word modifica- noncrossing, where graph $G$ is non- tion to sentence generation in natural crossing if there exists a linear arrange- language processing, which motivates ment of vertices so that no pair of edges 156 us to study this problem. are undirected and may have multiple This problem is a specialization of edges. We also assume that a path the book embedding[12] in the sense denotes a simple path throughout this that this problem asks if the given paper. CREW PRAM (see e.g., [5]) graph has page number one, i.e., a is adopted as a parallel computation graph can be embedded in a single model. page. In general, the book embedding is hard: it is NP-complete to tell if a 2 Forbidden Sub- planar graph can be embedded in two graph Characterization of a pages [3]. -
DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted By
DISSERTATION GENERALIZED BOOK EMBEDDINGS Submitted by Shannon Brod Overbay Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy Colorado State University Fort Collins, Colorado Summer 1998 COLORADO STATE UNIVERSITY May 13, 1998 WE HEREBY RECOMMEND THAT THE DISSERTATION PREPARED UNDER OUR SUPERVISION BY SHANNON BROD OVERBAY ENTITLED GENERALIZED BOOK EMBEDDINGS BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSO- PHY. Committee on Graduate Work Adviser Department Head ii ABSTRACT OF DISSERTATION GENERALIZED BOOK EMBEDDINGS An n-book is formed by joining n distinct half-planes, called pages, together at a line in 3-space, called the spine. The book thickness bt(G) of a graph G is the smallest number of pages needed to embed G in a book so that the vertices lie on the spine and each edge lies on a single page in such a way that no two edges cross each other or the spine. In the first chapter, we provide background material on book embeddings of graphs and preview our results on several related problems. In the second chapter, we use a theorem of Bernhart and Kainen and a result of Whitney to present a large class of two-page embeddable planar graphs. In particular, we prove that a graph G that can be drawn in the plane so that G has no triangles other than faces can be embedded in a two-page book. The discussion of planar graphs continues in the third chapter where we define a book with a tree-spine.