# Graph Theory. 1.1 Graph: Prepared by M. Aurangzeb Updated: Monday, March 15, 2010

Save this PDF as:

Size: px
Start display at page:

Download "Graph Theory. 1.1 Graph: Prepared by M. Aurangzeb Updated: Monday, March 15, 2010"

## Transcription

1 Prepared by M. Aurangzeb Updated: Monday, March 15, 2010 Graph Theory Graph theory is the branch of mathematics which deals with entities and their mutual relationships. The entities are represented by nodes or vertices and the existence of the relationship between nodes is represented as edges between/among the nodes. These nodes and relationships could be of a variety of nature like the nodes could be humans and edges could be the friendship between any two of them, nodes could be army officers and edges could be chain of command between them, nodes could be various destinations and edges could be roads between two of them, nodes could be cells in a cellular network and edges between two nodes could represent that they are adjacent, nodes could be atoms of Carbon, Oxygen and Hydrogen and edges representing covalent bonds between them, nodes could be critical points in a piping systems and edges could be pipes between them, nodes could be electrical components and edges could be connections between them, nodes could be states of a Markov process and edges could be possible transition between them and nodes could be various agents in a distributed system and edges could be the flow of information between them. These examples cover a wide range of walks of life including computer science, telecommunication, organic chemistry, distributed control systems and many others. Depending upon the application the nature of edges could be different. The edges could be simple non directional edges as incase of friendship between two persons and covalent bond between two atoms, they could be directional edges as in case of chain of command, there could be self edges as in case of Markov chains, there could be multiple edges between nodes as incase of two different roads between two destinations, there could be weighted edges as in case of road between two destinations weighted by the distance between two destinations or there could be the concept of hyper edges as incase of friendship groups. In these applications graph theory is used to address an endless list of problems. Just to name a few of them are shortest distance between two destinations, routing in computer networks, allocation of frequencies to cellular networks, determination of organic polymers, determination of minimum points to isolate two geographic regions, determination of flow between two nodes, analysis of electrical networks, design and analysis of data structure, distributed algorithms, parallel processing, load balancing and consensus in distributed networks. Depending upon the application, the graph theory deals with the various types of graphs like undirected, directed, simple, weighted and hyper graphs. To begin with let s choose the simple finite graphs and come to the formal definition of a simple graph, these simple graphs would be called graphs in our discussion. 1.1 Graph: A graph is an ordered pair G = (V,E) such that E [V] 2 where [V] k is the set containing all the subsets of V with k elements, these subsets of V are call k subsets of V. The cardinality of [V] k is represented as [V] k V and equal to C where V is the cardinality of V called the order of a graph and will be k represented by n in our discussion. Connecting this definition with the applications as discussed earlier, 1

2 V is called the set of vertices/nodes and E is called the set of edges. Consequently the elements of V are called vertices of the graph and usually denoted by v and the elements of E are called edges of the graph and usually denoted by e. Also referring to a graph G its order can also be represented as G (= V ) and its number of edges can also be represented as G (= E ). It must be noted here that a simple graph may not contain self loops, multiple edges. Also all their edges are undirected, connecting two vertices and do not have any weight. A vertex v is incident with an edge e if v e or we may say v is an end vertex of the edge e, an edge connects its two end vertices, and e is said to be an edge at v. If X and Y are two subsets of V then E(X,Y) represents the subset of E incident in both X and Y. Two vertices are neighbors/adjacent in G if there is an edge between them. Similarly two edges are adjacent if they have a common vertex. On contrary a set of vertices or edges is call independent of if they are pair wise non adjacent. Two vertices with an edge between them are called neighbors of each other while two vertices reachable from one another through a sequence of vertices and edges are called connected. A graph is called connected if every node of the graph is connected to every other node. A graph which is not connected is called a disconnected graph. A connected graph is said to have one component while a disconnected graph is the union of more than one connected components. The number of edges incident on a node v is called its degree and denoted by d(v). The mean degree of a node in a graph is denoted by d(g) and given by 1 dg ( ) dv ( ) V vv Also the minimum of all the degrees taken over all the vertices is called minimum degree of the graph and denoted by δ(g). Similarly maximum of all the degrees taken over all the vertices is called maximum degree of the graph and denoted by Δ(G). These definitions directly imply the following inequality ( G) d( G) ( G) Now let s come to a pretty straightforward but a famous result in graph theory known as Handshaking Lemma. Lemma (The Handshaking Lemma) The sum of degrees of all the nodes is equal to the number of edges in a graph. In its mathematical form we may write it as vv dv () 2 E If we sum all the degrees in a graph then every edge is counted twice and this completes our proof to this lemma. 2

3 Another entity related to graphs is number of edges of G per vertex and denoted by ε(g), mathematically we may write 1 1 ( G) E / V d( v) d( G) 2 V 2 The above equations are written with the help of the handshaking Lemma and the previously defined entities. There are some interesting results which are directly followed from the preliminaries. Lemma The number of vertices of odd degree is even. From the Handshaking Lemma sum of degrees of all the vertices are even. It is not possible unless number of vertices with odd degree is even. Lemma In every graph with more than 1 vertex there are at least two nodes with the same degree. If there are two or more vertices with zero degree (isolated vertices) we are done, otherwise there is a connected component in the graph with k (1< k n) vertices. In this connected component there is a choice for each of k vertices to have any degree from 1 to k 1. And by Pigeon Hole Principle at least two vertices must have the same degree. Lemma If δg n 2/2 n1 then the graph is connected. Let the graph is not connected and it has k (k>1) components. Under the given condition each component must be having more than n/2 nodes n>k (n/2). This is not possible for any value of k. 1.2 More on Graphs There are some special types of graphs we refer most often. The most famous of these is a complete graph. A graph G is called a complete graph if EG V 2, and represented as K n. If all the vertices of a graph are of the same degree then such a graph is called k regular. A graph G is called a bipartite graph if VG can be partitioned in two sets X and Y such as there is no edge whose both of the edges are in the same partition. We can similarly define an r partite graph. The notion can be further extended to a complete r partite graph. A complete r partite graph contains all the possible r i 1 edges for a given partition, if VG ( ) V and i V V then the corresponding r partite j j l l graph can be represented as K. A complete bipartite graph K 1, n 1 is called a star and V1, V2,..., Vr vv 3

4 represented as S n 1. A graph is called a chain/path if all of its vertices can be ordered in such a way that they are connected iff they are consecutive in the order. The graph obtained by adding an edge between two vertices of degree 1 in a chain is called a cycle and represented as C n. This is probably a suitable point where we may formally define a path within a graph. A sequence of distinct vertices stating from a vertex u to another vertex v such as all the consecutive vertices are adjacent in G is called a path xpy from u to v within G. If there exists an edge e u,v then xpv U e is called a cycle within G. A graph having a cycle as a subgraph is called a cyclic graph. While an acyclic graph is called a forest and an acyclic connected graph is called a tree. Like path there is another thing defined is graphs and it is a walk. A walk is like a path with permission of repetition of vertices. A graph is called a planner graph if it can be embedded in a plane in such a way that its edges intersect only at its vertices. A planer graph divides the plane in several parts known as faces. In these faces one is always unbounded and extending beyond the vertices and edges of the graph is called external face and also counted among the faces of the planer graph. The notion of sub graph is very similar to the notion of subsets except that in case of graphs it must be extended to both the set of vertices and the set of edges. There are some interesting refinements in this notion. The first one is known as induced subgraph. If G G and EG contain all the edges xy EGwith x, y VG then G is called an induces subgraph of G. We also say that VG V induces or spans G in G and write it as G GV. The second one is spanning subgraph, if G G and VG VG then G is known as a spanning subgraph of G. In particular if the spanning subgraph is a tree then it is called a spanning tree of the graph. The third one is the subgraph obtained by deleting an edge e of the graph G and written as G\e. The fourth one is obtained by contracting an edge e of the graph G and written as G/e. Please note that the direction of slash makes a difference in these notations also the detailed discussion on the last two will follow in the later part of this text. The fifth one is due to the deletion of a vertex v of G, written as G v which is obtained from G by deleting the vertex v and edges incident on it. Several relationships could be defined in graphs; some of them are mentioned here. The notions of union and intersection in a graph are very much similar to the respective notions in the set theory with the only detail that for graphs the respective operations must be extended to the set of vertices and edges. Next, for a graph GV,E, its compliment is denoted by G V, E where E V 2 E. Two graphs G and G are said to be isomorphic if there exists a bijection φ:vg VG such as xy EG φx φy EG and we write it as GG. For a graph G, there is another graph LG associated with it which is called line graph of G and obtained from G by presenting all the edges of G as the vertices in LG with edges between them iff they were incident on a common vertex in G. Similarly if we have a planer graph G then another graph G* obtained by expressing each face of G by a node in G* with edges between them iff corresponding faces in G had common edge. It must be noted that here we have an edge in G* against each edge in G which may cause to generate a multi graph. G * is called dual of G for the particular embedding. Our knowledge about graphs up to this extent allows us to work with a little involved results in the graph theory. 4

5 Proposition Every graph G with G0 has a subgraph H with δh ε H ε G Before going into the proof of the proposition we need to examine k regular graphs. For these graphs k0, Gk and ε Gk/2, this shows that the proposition holds for such graphs. Now if the graph G does not have any vertex whose degree is lesser than or equal to ε G the proposition already holds. However if there is a vertex with degree lesser than or equal to ε G we may plan to delete such vertex to get a new graph G 1. For this graph G 1 G 1 and G 1 G e d, where e d is the number of edges deleted which due to the selection of the vertex is lesser than or equal to ε G. This will cause a total reduction of 2e d in the sum of degrees of all the nodes in G 1. We may say G ( G) G and G G e 1 d ( G1 ) whereed ( G) G1 G 1 And it can be seen readily that under the given condition ( G1 ) ( G). The process is carried on recursively getting G i1 from G i with ( Gi 1) ( Gi) every subsequent subgraph having greater value of ( G i ). The process is continued till we get the subgraph H G i with δh ε H. Needless to mention that the process deemed to terminate before the trivial graph because and regular graph including K2 has already been seen to obey the proposition. The above result gives us a procedure to find well connected nodes in a graph which are of great interest in many applications. The next result we are going to discuss relates to the G with the length of path and cycles within the graph. Proposition Every graph G contains a path of length δg and a cycle of length at least δg1 provided δg 2. Let us consider a maximal path Pv 1, v 2,,v k within the graph. As the path is maximal so all the neighbors of v k must be present in P, and this completes the first part of the proposition. For δg 2 there must be at least one neighbor of v k other than v k 1 in P. Suppose that v i is the neighbor of v k with the least index in P, and we have just found a cycle v i, v i1,,v k, v i with length greater than or equal to δg1. To have further enjoyment we need to know a few more things about graphs. The first one to come is the girth of a graph which the length of minimum cycle presents in the graph, it is denoted by 5

7 V 1Δ Δ 1 radg 1/ Δ 2 V Δ Δ 1 radg / Δ 2 1 Δ/ Δ 2 V Δ Δ 1 radg / Δ 2 and for kradg V Δ Δ 1 K / Δ 2 Lemma For a graph G with minimum degree 3 and girth g has at least n o, g vertices where g1 2 (( 1) 1) 1 when g is odd 2 no (, g) g 2 (( 1) 1) 2 when g is even 2 Let g be odd. Let us start from any vertex v. It needs to have at least neighbors. Now each of these neighbors must have at least 1 distinct neighbors. And starting from v this process will go on for at least g 1/2 steps. We can thus write the following inequality V g 3/2 For g 3, this is the least odd value for the girdth V 1 1 g 1/2 1/ 2 V n o, g And we are done for the odd girth. Now let g be even. Now starting from two adjacent vertices u and v, each of them needs to have at least 1 neighbors other than u and v, and similarly each of these neighbors must have at least 1 distinct neighbors. However, this time starting from v this process will go on for at least g 2/2 steps. We can thus write the following inequality V g 2/2 For g 4, this is the least even value for the girth V 2 1 g/2 1/ 2 V n o, g And we are done Now we come to the following proposition which gives us a very important result for the connected graphs. 7

8 Proposition The vertices of a connected graph can always be enumerated as v 1, v 2, v N such as G i Gv 1,v 2, v i is connected for every i 1,2, N. Let s start with any vertex v 1 within the enumeration. As a single to graph is trivially connected the statement is true for i1. Now suppose that G k Gv 1,v 2, v k is connected. Now take any vertex v which has not been enumerated yet. Since the graph is connected, starting from v we can always reach an enumerated vertex, let v j is the first one we reach following any suitable path. We enumerate the second last vertex as v k1 in the path v~v j, and we are inductively done as G k1 Gv 1,v 2, v k,v k1 is connected. As defined earlier an acyclic connected graph is a tree. Here are a few facts about the trees to further elaborate its structure. Proposition The following statements are equivalent. i. T is a tree ii. Any two vertices in T are connected by a unique path iii. T is minimally edge connected iv. T is maximally acyclic Let T be a tree. T is connected and acyclic Any two vertices of T are connected by a unique path; if there are more than one path T would no longer acyclic. iii Let any two vertices in T are connected by a unique path Deletion of any edge would disconnect at least two vertices T is minimally edge connected iiiii Let T be minimally edge connected Deletion of any on edge will render the graph disconnected. T is acyclic Also any two vertices x, y are connected in T 8

9 Addition of edge xy would create a cycle T is maximally acyclic iiiiv Lastly let T is maximally acyclic Addition of any xy edge would create a cycle Any two vertices x and y are connected in T T being connected and acyclic is a tree ivi This completes our proof. Proposition The vertices of a tree T can always be enumerated as v 1, v 2, v N such as v i i=2 N has a unique neighbor in v 1,v 2, v i 1. From the Proposition its well clear that vertices of any connected graph can be enumerated as v 1, v 2, v N such as G i Gv 1,v 2, v i is connected i=1,2 N. G i Tv 1,v 2, v i 1 and G i Tv 1,v 2, v i i=2 N are both connected v i is has at least one neighbor in v 1,v 2, v i 1 Now for the uniqueness suppose that v i has two neighbors in v 1,v 2, v i 1, let they be v k and v kj. By Proposition v k and v kj are connected. v k and v kj and v i will make a loop, which is in contradiction to the fact that T is a tree v i has a unique neighbor in v 1,v 2, v i 1 i=1,2 N Proposition A connected graph with N vertices is a tree if and only if it has N 1 edges. Let s have a tree T with N vertices. By the Proposition the vertices of T can be enlisted as v 1, v 2, v N such as each vertex in the list has a unique neighbor among the preceding vertices. Thus counting all the edges of a vertex to its preceding vertices these would be N 1 as the first one would not be having a preceding vertex. Conversely suppose that T is a connected graph with N 1 edges. We use induction to establish this part. For N2 the result is trivially true. Now let the result is true for Nk i.e. T, a connected graph with k vertices and k 1 edges is a tree. Now we add a vertex to this graph so that it remains a tree. 9

10 For this purpose we my either connect to an existing vertex or insert within an existing edge. In both the case only one edge would be added and we are done. The next proposition gives us an important result for bipartite graphs. Proposition A graph is bipartite if and only if it has no odd cycles. Let s have a bipartite graph. It can be readily seen that there could not be an odd cycle. Conversely let the graph has no odd cycle. To prove that it is bipartite we have to show that all of its components are bipartite. Thus without loss of generality we suppose that the graph is connected. Now enumerate the nodes as we did in Proposition but this time with an additional step; each time we add a node we assign it a color either red or blue. With the first node assigned with red, we keep on assigning them color in such a way that each newly added node has a color different from it s already enumerated neighbors. Our claim is that it is always possible. It is trivial to see that under given condition of the absence of odd cycle it is always possible for some small number of nodes. Now inductively suppose that it is possible for us to enumerate/color k nodes in this way i.e. we have an enumeration v 1, v 2 v k such as G k v 1, v 2 v k is connected and with every node having a color different from its preceding neighbors. Now every node with the same different color in the enumeration is even odd hops apart. Now if we add a node v k1 in this enumeration then it will have all the neighbors in G k1 v 1, v 2 v k, v k1 having same color; else it would have odd cycles. So we may assign it with a color different from its neighbors in G k1. And placing the nodes with same color on one side we have shown a bipartition of the graph. Now we come to probably the most famous notion in graph theory; Euler Tour. A closed walk in graph is called Euler tour if it contains every edge exactly once. A graph where it is possible to have an Euler tour is called Eulerian. Theorem (Euler) A connected graph is Eulerian if and only if all of its vertices are of even degree. The necessity of an Eulerian graph to have all its vertices with even degree is evident from the fact that for each incoming edge to a vertex we need to have an outgoing edge to complete the Eulerian tour. Conversely suppose that all the vertices of a connected graph have even degree. Now starting from any one node v 1 we would always come back to the same node as for every edge we enter into a node there would always be an unvisited edge to go out of that vertex except possibly for the initial vertex v 1. Now if we have already visited all the edges we are done. If not, then for our connected graph there need to have an already visited vertex v 2 with an unvisited edge. Following this edge, owing to the same reason as established earlier, we are again bound to come back to v 2. Continuing in the same way we would surely end at some vertex v k, when we have visited all the edges. Now the sequence v 1 ~v 2 ~ ~v k 1 ~v k ~v k ~v k 1 ~ v 2 ~v 1 is our desired Eulerian tour. 10

11 Now we come to an important concept in graph theory, known as degree sequence. If we arrange the degrees of all the vertices of a graph in non increasing order then this sequence is known as the degree sequence of the graph. A question naturally arises that whether all the non increasing finite sequence of nonnegative integers corresponds to a graph. The answer to this question is No. A non increasing finite sequence of nonnegative integers is known as graphic if there is a graph with the given sequence as its degree sequence. There is a necessary and sufficient condition for a sequence to be graphic. Theorem A finite sequence of nonnegative integers d d 1,d 2, d N with d 1 d 2 d N is graphic if and only if N k N d is even and d k( k 1) min( k, d ) k N 1 i i i i1 i1 ik1 Let the given sequence of nonnegative integers d d 1,d 2, d N with d 1 d 2 d N is graphic There is a graph G with vertices having degrees d 1,d 2, d N By using Lemma Hand Shaking Lemma sum of the degrees of all the vertices will be even. This proves the first part of the necessary condition. Now for the second part of the necessary condition k i1 k i1 k d Sum of all thedegree of G [ d, d,... d ] Number of edges from {v,v,...v } to V-{v,v,...v } i k 1 2 k 1 2 k 1 2 k d 2 k( k1) / 2 Number of edges from {v,v,...v } to V-{v,v,...v } i 1 2 k 1 2 k d k( k1) min( k, d ) i i1 ik1 N And this establishes the necessary condition. i Now for the sufficiency suppose that N k N d is even and d k( k 1) min( k, d ) k N 1 i i i i1 i1 ik1 We will devise a method to make a graph corresponding to the given degree sequence under the given conditions. Work in progress M. Aurangzeb

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction

### IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### The Clar Structure of Fullerenes

The Clar Structure of Fullerenes Liz Hartung Massachusetts College of Liberal Arts June 12, 2013 Liz Hartung (Massachusetts College of Liberal Arts) The Clar Structure of Fullerenes June 12, 2013 1 / 25

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius

### Course on Social Network Analysis Graphs and Networks

Course on Social Network Analysis Graphs and Networks Vladimir Batagelj University of Ljubljana Slovenia V. Batagelj: Social Network Analysis / Graphs and Networks 1 Outline 1 Graph...............................

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

### UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE

UPPER BOUNDS ON THE L(2, 1)-LABELING NUMBER OF GRAPHS WITH MAXIMUM DEGREE ANDREW LUM ADVISOR: DAVID GUICHARD ABSTRACT. L(2,1)-labeling was first defined by Jerrold Griggs [Gr, 1992] as a way to use graphs

### Solutions to Homework 6

Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Ph.D. Thesis. Judit Nagy-György. Supervisor: Péter Hajnal Associate Professor

Online algorithms for combinatorial problems Ph.D. Thesis by Judit Nagy-György Supervisor: Péter Hajnal Associate Professor Doctoral School in Mathematics and Computer Science University of Szeged Bolyai

### SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

SHOT YLE OVES OF PHS WITH MINIMUM DEEE THEE TOMÁŠ KISE, DNIEL KÁL, END LIDIKÝ, PVEL NEJEDLÝ OET ŠÁML, ND bstract. The Shortest ycle over onjecture of lon and Tarsi asserts that the edges of every bridgeless

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### On the independence number of graphs with maximum degree 3

On the independence number of graphs with maximum degree 3 Iyad A. Kanj Fenghui Zhang Abstract Let G be an undirected graph with maximum degree at most 3 such that G does not contain any of the three graphs

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### Cycles and clique-minors in expanders

Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

### MATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### Simple Graphs Degrees, Isomorphism, Paths

Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### Disjoint Compatible Geometric Matchings

Disjoint Compatible Geometric Matchings Mashhood Ishaque Diane L. Souvaine Csaba D. Tóth Abstract We prove that for every even set of n pairwise disjoint line segments in the plane in general position,

### SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE 2012 SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH (M.Sc., SFU, Russia) A THESIS

### Small Maximal Independent Sets and Faster Exact Graph Coloring

Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### (Refer Slide Time: 01.26)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 27 Pigeonhole Principle In the next few lectures

### Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

### Graph Theory: Penn State Math 485 Lecture Notes. Christopher Griffin 2011-2012

Graph Theory: Penn State Math 485 Lecture Notes Version 1.4..1 Christopher Griffin 011-01 Licensed under a Creative Commons Attribution-Noncommercial-Share Alike.0 United States License With Contributions

### Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. #-approximation algorithm.

Approximation Algorithms 11 Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of three

### Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### ! Solve problem to optimality. ! Solve problem in poly-time. ! Solve arbitrary instances of the problem. !-approximation algorithm.

Approximation Algorithms Chapter Approximation Algorithms Q Suppose I need to solve an NP-hard problem What should I do? A Theory says you're unlikely to find a poly-time algorithm Must sacrifice one of

### Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

### most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia

Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle

### A Study of Sufficient Conditions for Hamiltonian Cycles

DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph

### A threshold for the Maker-Breaker clique game

A threshold for the Maker-Breaker clique game Tobias Müller Miloš Stojaković October 7, 01 Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n, p. In this game,

### Determination of the normalization level of database schemas through equivalence classes of attributes

Computer Science Journal of Moldova, vol.17, no.2(50), 2009 Determination of the normalization level of database schemas through equivalence classes of attributes Cotelea Vitalie Abstract In this paper,

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

### 6.852: Distributed Algorithms Fall, 2009. Class 2

.8: Distributed Algorithms Fall, 009 Class Today s plan Leader election in a synchronous ring: Lower bound for comparison-based algorithms. Basic computation in general synchronous networks: Leader election

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### CIS 700: algorithms for Big Data

CIS 700: algorithms for Big Data Lecture 6: Graph Sketching Slides at http://grigory.us/big-data-class.html Grigory Yaroslavtsev http://grigory.us Sketching Graphs? We know how to sketch vectors: v Mv

### Non-Separable Detachments of Graphs

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2001-12. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### Even Faster Algorithm for Set Splitting!

Even Faster Algorithm for Set Splitting! Daniel Lokshtanov Saket Saurabh Abstract In the p-set Splitting problem we are given a universe U, a family F of subsets of U and a positive integer k and the objective

### HOLES 5.1. INTRODUCTION

HOLES 5.1. INTRODUCTION One of the major open problems in the field of art gallery theorems is to establish a theorem for polygons with holes. A polygon with holes is a polygon P enclosing several other

### Discrete Mathematics. Hans Cuypers. October 11, 2007

Hans Cuypers October 11, 2007 1 Contents 1. Relations 4 1.1. Binary relations................................ 4 1.2. Equivalence relations............................. 6 1.3. Relations and Directed Graphs.......................

### Linear Programming I

Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902 http://www.ms.uky.edu/~mschreffler Different Graphs, Similar Properties

### Network File Storage with Graceful Performance Degradation

Network File Storage with Graceful Performance Degradation ANXIAO (ANDREW) JIANG California Institute of Technology and JEHOSHUA BRUCK California Institute of Technology A file storage scheme is proposed

### Multi-layer Structure of Data Center Based on Steiner Triple System

Journal of Computational Information Systems 9: 11 (2013) 4371 4378 Available at http://www.jofcis.com Multi-layer Structure of Data Center Based on Steiner Triple System Jianfei ZHANG 1, Zhiyi FANG 1,

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### Diversity Coloring for Distributed Data Storage in Networks 1

Diversity Coloring for Distributed Data Storage in Networks 1 Anxiao (Andrew) Jiang and Jehoshua Bruck California Institute of Technology Pasadena, CA 9115, U.S.A. {jax, bruck}@paradise.caltech.edu Abstract

Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NP-hard problem. What should I do? A. Theory says you're unlikely to find a poly-time algorithm. Must sacrifice one

### The Basics of Graphical Models

The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures

### Near Optimal Solutions

Near Optimal Solutions Many important optimization problems are lacking efficient solutions. NP-Complete problems unlikely to have polynomial time solutions. Good heuristics important for such problems.

### Codes for Network Switches

Codes for Network Switches Zhiying Wang, Omer Shaked, Yuval Cassuto, and Jehoshua Bruck Electrical Engineering Department, California Institute of Technology, Pasadena, CA 91125, USA Electrical Engineering

### 3. Linear Programming and Polyhedral Combinatorics

Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles

Short Cycles make W-hard problems hard: FPT algorithms for W-hard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman

### Practical Graph Mining with R. 5. Link Analysis

Practical Graph Mining with R 5. Link Analysis Outline Link Analysis Concepts Metrics for Analyzing Networks PageRank HITS Link Prediction 2 Link Analysis Concepts Link A relationship between two entities

### Global secure sets of trees and grid-like graphs. Yiu Yu Ho

Global secure sets of trees and grid-like graphs by Yiu Yu Ho B.S. University of Central Florida, 2006 M.S. University of Central Florida, 2010 A dissertation submitted in partial fulfillment of the requirements

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan