SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

Size: px
Start display at page:

Download "SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH"

Transcription

1 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 and B-diameter of a block in a graph. An algorithm is given to find the block center of the graph. It is proved that block center of any graph is B-complete. Later six new graphs are defined viz - semitotal block cutvertex graph, total block cutvertex graph, semitotal block vertex graph, total block vertex graph, semitotal block vertex edge graph and total block vertex edge graph arising from the given graph. Expressions for number of edges in the newly defined graphs are derived.

2 3.1 Introduction For standard terminologies we refer F. Harary [1]. Distance plays a prominent role in the study of Graph Theory. For any two vertices a, b in G the distance d ( a, b) is the length of a shortest path between a and b. Let x, y X, a V and x ( u, v) then the distance between the vertex a and an edge x, is defined as d( x, a) min{ d( u, a), d( v, a)} and the distance between the two edges d( x, y) min{ d( y, u), d( y, v)}. The eccentricity of a vertex v V(G) is defined as. Then diameter and radius. For any graph G,. The set of vertices with minimum eccentricity is called center C (G) of the graph. For any tree T the center consists of a single vertex or two adjacent vertices. Using distances, several problems are posed in graph theory such as Chinese postman problem and shortest distance problem etc. Many well known algorithms like Dijikstra s algorithm, Kruskal s algorithm and Prim s algorithm have been generated using the distance concepts. Similar to distance in graphs, block distance in graphs are defined. 3.2 B-distance between two blocks B-paths are defined in Chapter 2. We recall that the length of a B-path is the number of cutvertices in it. Definition For any two blocks b 1, b 2 B(G), the B-distance d(b 1, b 2 ) is the length of the B-path from b 1 to b 2. Further, for any b 1, b 2, b 3 B(G) the block distance has the following properties. 42

3 (i) d(b 1, b 2 ) 0 and d(b 1, b 2 ) = 0 if and only if b 1 = b 2. (ii) d(b 1, b 2 ) = d(b 2, b 1 ) (iii) d(b 1, b 2 ) d(b 1, b 3 ) + d(b 3, b 2 ) - Triangular inequality. Therefore B-distance is a metric. Using B-distance the central tendencies of a block, like B-radius and B-diameter of a block in a graph can also be defined. Definition The B-eccentricity of a block h B(G) is defined as. Then B-diameter and B-radius. The set of blocks with minimum eccentricity is called B-center C b (G) of the graph. Immediately these parameters yield a lower bound for the usual diameter d(g) and radius r(g) of a graph. Hence we have and. Proposition Let G be a graph with B-diameter d b (G) and B-radius r b (G). Let and be the block graph and cutvertex graph of G. Then (3.1) ( ) (3.2) ( ) (3.3) ( ) (3.4) ( ) (3.5) Proof. Equation (3.1) follows from the fact that. To prove (3.2). Consider the diametrical B-path in G. 43

4 Then = number of cutvertices in the diametrical B-path in G = number of blocks in the diametrical B-path in B G (G) = number of cutvertices in the diametrical B-path in B G (G) + 1 = ( ). Since B G (B G (G)) = C G (G) equation (3.3) follows from (3.2). To prove (3.4). Again from (3.1) and (3.2), we have ( ( ) ) ( ) ( ) as desired. Also from (3.3) we have, ( ). Hence ( ) ( ) which yields (3.5). Recall that a skeleton of G is obtained by deleting all the pendant blocks from G. An i th skeleton S i (G) is obtained iteratively by getting skeleton of G, i times. Therefore G = S 0 (G). Definition If a vertex v is a cutvertex both in G and S 1 (G) then v is a non-end-cutvertex of G, otherwise v is an end-cutvertex of G. Proposition (i) (ii) If G has atleast one cutvertex then there exists atleast two pendant blocks. If G has atleast two cutvertices then there exists atleast two end-cutvertices. Proof. Consider the diametrical B-path in G. Then the two blocks with maximum B-eccentricity are the required two pendant blocks and the cutvertex incident on these two blocks are the required two end-cutvertices. Proposition For any graph G with m blocks the B-center C b (G) is B-complete. Proof. Case 1. Suppose m = 1. Then d b (G) = r b (G) = 0. Hence C b (G) = B 1 and the result follows. 44

5 Case 2. Suppose m >1 and G = B m Then d b (G) = r b (G) = 1. Hence C b (G) = B m and the result follows. Case 3. Suppose m >1 and G is not a B-complete graph. Then G has atleast two cutvertices. Then by Proposition there exists atleast two pendant blocks. Now obtain the skeleton S 1 (G). Then all the blocks are at B-distance one less than the B- distance in the original graph. Also the blocks which were in the B-center in the original graph remain same in the new graph. Hence C b (G) = C b (S 1 (G)). Let k 1 be the number of blocks in S 1 (G). Check whether S 1 (G) is B-complete and if so, from case 1 and case 2, the result follows and we stop. If S 1 (G) is not B-complete, we get S 2 (G). The process of getting skeleton S i +1 (G) from S i (G) is repeated, if and only if S i (G) is not B-complete. After say n steps we get S n (G) which is B-complete with k n blocks and C b (G) = C b (S n (G)) = S n (G) which is B-complete. Then by case 1 and case 2, the result follows. This completes the proof. From the above proof it is possible to get an algorithm to find B-centre of G which eventually gives the B-radius of G. Note that if we take skeleton of G, i times implies that block radius of G is i or i + 1 according as B-center is B 1 or B m where m >1. The algorithm proceeds as follows. Algorithm i = 0 count = m k = count begin while (S i (G) B k ) S i (G) = S i+1 (G) i = i + 1 count = number of blocks in S i+1 (G) 45

6 end k = count end while C b (G) = S i (G) if k = 1 block radius of G = i else block radius of G = i+1 end if 3.3 Semitotal-block-cutvertex graph and total-block-cutvertex graph. The line graph L(G) of a graph G is a graph with vertex set X(G) and any two vertices in L(G) are adjacent if and only if the corresponding lines are adjacent in G. The block graphs are studied in [70 and 72]. The edges and vertices of G are called its members. Semitotal graph t(g) is a graph as defined by Sampathkumar and Chikodimath [73] with vertex set X(G) V(G) and any two vertices in t (G) are adjacent if and only if the corresponding vertices are adjacent in G or the corresponding members are incident. The total graph T(G) introduced by M. Behzad [74] is a graph with vertex set X(G) V(G) and two vertices in T(G) are adjacent if and only if the corresponding members are adjacent or incident in G. V.R.Kulli [75] defined and studied the properties of semitotal- block graph and total-block graphs. The vertices and blocks of a graph are called its members. Semitotal-block graph T b (G) of a graph G is a graph with vertex set B(G) V(G) and two vertices are adjacent if and only if the corresponding vertices are adjacent in G or corresponding members are incident. The total block graph T B (G) has vertex set B(G) V(G) and two elements in T B (G) are adjacent if and only if they are adjacent or incident to each other in G. Similar to block cutvertex graph, V.R.Kulli [75] defined the block 46

7 vertex tree of a graph G, denoted as bp(g) which is a tree with vertex set B(G) V(G) and a vertex v V (G) and a block b B (G) are adjacent if and only if v is incident with the block b. Motivated by these definitions we now define two new graphs arising from the given graph. Definition The blocks and cutvertices of a graph are called its members. Semitotal block cutvertex graph T bc (G) of a graph G is a graph with vertex set B(G) C(G) and any two vertices in T bc (G) are adjacent if and only if the corresponding cutvertices are adjacent or the corresponding members are incident. It is immediate that T bc (G) = BC(G) C G (G). Definition The total block cutvertex graph T BC (G) of a graph G is a graph with vertex set B(G) C(G) and any two vertices in T BC (G) are adjacent if and only if the corresponding members are adjacent or incident. Again we note that T BC (G) = BC(G) C G (G) B G (G). It is important to recall the following definitions defined in Chapter 2 to prove the next proposition. A block b is a pendant block if it is incident with only one cutvertex, otherwise b is a nonpedant block. Let B P (G) and B NP (G) denote the set of all pendant and nonpendant blocks of G respectively. Let m P = B P (G) and m NP = B NP (G). B NP (G) B P (G) =B(G) and B NP (G) B P (G) =, hence m = m p + m NP. Proposition For any graph G, with m NP pendant blocks and n cutvertices, (3.6) ( ) ( ) (3.7) 47

8 Proof. From Proposition 2.4.2, we have. Since each pendant block (has only one cutvertex) contribute one to the sum of cutvertex degrees, we have. Then Now, ( ) ( ) ( ) ( ). Hence the proof. An expression for the number of edges in the semitotal and total block cutvertex graph is derived in the next Theorem. Henceforth, by we mean the number of edges in. Theorem Let G be a graph with m blocks and n cutvertices. Let p bc and q bc denote the number of vertices and number of edges in T bc (G). Let p BC and q BC denote the number of vertices and number of edges in T BC (G). Then (i) p bc = m + n = p BC (3.8) (ii) ( ( ) ) (3.9) (iii) [ ( ) ( ) ] (3.10) Proof. Since the vertex set of T bc (G) and T BC (G) being the disjoint union of B(G) and C(G) we have p bc = p BC = B(G) + C(G) = m + n. To prove equation (3.9). Since every cutvertex yields d vb (c) edges in BC(G), we have q(bc(g)) =. Further, since all the cutvertices incident to a 48

9 nonpendant block are mutually adjacent, every nonpendant block h yields ( ) edges in C G (G). Hence ( ) ( ). Then, ( ) ( ) ( since T bc (G) = BC(G) C G (G) ( ) ( ) ( ) ( ) (using Proposition 3.3.3) ( ) ( ( ) ) (since m- m NP = m p ). ( ( ) ) ( ( ) ). To prove equation (3.10). Since all the blocks incident to a cutvertex c are mutually adjacent, every cutvertex yields ( ) edges in B G (G). Hence ( ) ( ). Then, ( ) ( ) ( ) (since ) ( = ) ( ) [ ( ) ( ) ] 49

10 (using Proposition 3.3.3) [ ( ) ( ) ] (Since from Theorem 2.4.2) [ ( ) ( ) ] (since )). Example For the graph G in the Fig.3.3.6, ( ) ( ( ) ) which in agreement with the result obtained in the Theorem Similarly, [ ( ) ( ) ] which satisfies the result of the Theorem T T bc (G) BC (G) G Fig Semitotal and Total block- cutvertex graph of G 3.4 Semitotal-block-vertex graph and total-block-vertex graph Definition The blocks and vertices of a graph are called its members. Semitotal-blockvertex graph T bp (G) is a graph with vertex set V(G) B(G) and any two vertices in T bp (G) are adjacent if and only if the corresponding vertices are vv-adjacent or the corresponding members are incident. It is immediate that T bp (G) = b p (G) P G (G). Definition The total-block-vertex graph T BP (G) is a graph with vertex set V(G) B(G) and any two vertices in T BP (G) are adjacent if and only if the corresponding members are vv-adjacent or adjacent or incident. Again we note that T BP (G) = b p (G) P G (G) B G (G). 50

11 It is obtained an expression for the number of edges in the semitotal-block-vertex and total-block-vertex graphs in the next result. Theorem Let G be a (p, q) graph with m blocks and n cut-vertices. Let q bp and p BP denote the number of edges in T bp (G) and T BP (G) respectively. Then Proof. To prove equation (3.11). ( ( ) ) (3.11) [ ( ) ( ) ] (3.12) Since every vertex yields d bv (h) edges in b p (G), ( ). (from Theorem 2.4.3). Then, ( ) ( ) [ ( ) ] [ ( ) ]. To prove equation (3.12). ( ) ( ) ( ) [ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ] [ ( ) ( ) ] (using Theorem and Theorem 2.4.3) 51

12 Example For the graph G in the Fig.3.4.5, ( ( ) ) which in agreement with the result obtained in the Theorem Similarly, [ ( ) ( ) ] which satisfies the result obtained in Theorem G T bp (G) Fig Semitotal and total block- vertex graph of G T BP (G) 3.5 Semitotal-block-vertex-edge graph and total-block-vertex-edge graph Definition The blocks, vertices and edges of a graph are called its members. The Semitotal-block-vertex-edge graph T bpe (G) is a graph with vertex set V(G) B(G) X(G) and any two vertices in T bpe (G) are adjacent if the corresponding vertices are vv-adjacent or corresponding members are incident. Definition A vertex edge graph V e (G) is a bigraph with vertex set as V(G) X(G) and a vertex v V (G) and an edge x X(G) are adjacent in V e (G) if and only if v is incident on the edge x. 52

13 Definition A block edge graph b e (G) is a bigraph with vertex set as B(G) X(G) and a block b B (G) and an edge x X(G) are adjacent in b e (G) if and only if the edge x is incident on the block b. It is immediate that T bpe (G)= P G (G) b e (G) V e (G) b p (G). Definition The total-block-vertex-edge graph T BPE (G) is a graph with vertex set V(G) B(G) X(G) and any two vertices in T BPE (G) are adjacent if the corresponding members are vvadjacent or adjacent or incident. It is immediate that T BPE (G)= T bpe (G) B G (G) L(G). We note that the number of vertices in T BPE (G) and T bpe (G) is equal to p + q + m. graphs. In the next theorem we obtain an expression for number of edges in the above defined Theorem Let G be a (p, q) graph with m blocks and n cut-vertices. Let q bpe and p BPE denote the number of edges in T bpe (G) and T BPE (G) respectively. Then ( ( ) ) (3.13) [ ( ) ( ) ( ) ] (3.14) Proof. To prove (3.13). Since each edge is incident on two vertices, there are 2q edges in vertex edge graph V e (G). As every edge is incident on a unique block, there are q edges in block edge graph b e (G). Then, ( ) ( ) ( ) ( ) [ ( ) ] [ ( ) ] 53

14 To prove equation (3.14) We know that ( ) [ ( ) ] (see Harary [1]). Then, Example ( ) ( ) [ ( ) ] [ ( ) ( )] [ ( ) ] [ ( ) ( ) ( ) ] V 2 G V 1 T bpe (G) T BPE (G) Fig Semitotal and total block- vertex- edge graph of G For the graph in the Fig.3.5.7, ( ) ( ) ( ) ( ). Therefore Again ( ) ( ( ) ) which satisfies the result obtained in the Theorem Similarly, ( ) ( ) Also [ ( ) ( ) ]. Hence the Theorem is verified. 54

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

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.,

More information

On Integer Additive Set-Indexers of Graphs

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

More information

Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

More information

BOUNDARY EDGE DOMINATION IN GRAPHS

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

More information

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

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

More information

Connectivity and cuts

Connectivity and cuts Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

More information

Approximation Algorithms

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

More information

The Goldberg Rao Algorithm for the Maximum Flow Problem

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 }

More information

Analysis of Algorithms, I

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

More information

On the independence number of graphs with maximum degree 3

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

More information

2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines 2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

More information

GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

More information

Network (Tree) Topology Inference Based on Prüfer Sequence

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,

More information

CSC 373: Algorithm Design and Analysis Lecture 16

CSC 373: Algorithm Design and Analysis Lecture 16 CSC 373: Algorithm Design and Analysis Lecture 16 Allan Borodin February 25, 2013 Some materials are from Stephen Cook s IIT talk and Keven Wayne s slides. 1 / 17 Announcements and Outline Announcements

More information

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

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

More information

136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

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

More information

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree

Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Balloons, Cut-Edges, Matchings, and Total Domination in Regular Graphs of Odd Degree Suil O, Douglas B. West November 9, 2008; revised June 2, 2009 Abstract A balloon in a graph G is a maximal 2-edge-connected

More information

8.1 Min Degree Spanning Tree

8.1 Min Degree Spanning Tree CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

More information

A simple criterion on degree sequences of graphs

A simple criterion on degree sequences of graphs Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

More information

Cycles and clique-minors in expanders

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

More information

On end degrees and infinite cycles in locally finite graphs

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

More information

Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design

Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and Hyeong-Ah Choi Department of Electrical Engineering and Computer Science George Washington University Washington,

More information

On Some Vertex Degree Based Graph Invariants

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

More information

Special Classes of Divisor Cordial Graphs

Special Classes of Divisor Cordial Graphs International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

More information

GRAPHS AND ZERO-DIVISORS. In an algebra class, one uses the zero-factor property to solve polynomial equations.

GRAPHS AND ZERO-DIVISORS. In an algebra class, one uses the zero-factor property to solve polynomial equations. GRAPHS AND ZERO-DIVISORS M. AXTELL AND J. STICKLES In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x

More information

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

More information

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 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

More information

Euler Paths and Euler Circuits

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

More information

Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10

Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs

More information

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge,

The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, The Union-Find Problem Kruskal s algorithm for finding an MST presented us with a problem in data-structure design. As we looked at each edge, cheapest first, we had to determine whether its two endpoints

More information

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

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,

More information

Shortcut sets for plane Euclidean networks (Extended abstract) 1

Shortcut sets for plane Euclidean networks (Extended abstract) 1 Shortcut sets for plane Euclidean networks (Extended abstract) 1 J. Cáceres a D. Garijo b A. González b A. Márquez b M. L. Puertas a P. Ribeiro c a Departamento de Matemáticas, Universidad de Almería,

More information

Lecture 4: BK inequality 27th August and 6th September, 2007

Lecture 4: BK inequality 27th August and 6th September, 2007 CSL866: Percolation and Random Graphs IIT Delhi Amitabha Bagchi Scribe: Arindam Pal Lecture 4: BK inequality 27th August and 6th September, 2007 4. Preliminaries The FKG inequality allows us to lower bound

More information

Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion. Ernie Croot. February 3, 2010 Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

More information

MATHEMATICS Unit Decision 1

MATHEMATICS Unit Decision 1 General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Decision 1 MD01 Tuesday 15 January 2008 9.00 am to 10.30 am For this paper you must have: an 8-page answer

More information

Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

More information

On the k-path cover problem for cacti

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

More information

A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY

A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY International Journal of Pure and Applied Mathematics Volume 03 No. 05, 6-70 ISSN: 3-8080 (printed version); ISSN: 34-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5

More information

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. 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

More information

1 Approximating Set Cover

1 Approximating Set Cover CS 05: Algorithms (Grad) Feb 2-24, 2005 Approximating Set Cover. Definition An Instance (X, F ) of the set-covering problem consists of a finite set X and a family F of subset of X, such that every elemennt

More information

NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics

NP-complete? NP-hard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics NP-complete? NP-hard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer

More information

Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Degree-associated reconstruction parameters of complete multipartite graphs and their complements Degree-associated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertex-deleted subgraphofagraphgisacard.

More information

Cycles in a Graph Whose Lengths Differ by One or Two

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

More information

Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

More information

3. Equivalence Relations. Discussion

3. Equivalence Relations. Discussion 3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

More information

SHORT CYCLE COVERS OF GRAPHS WITH MINIMUM DEGREE THREE

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

More information

M-Degrees of Quadrangle-Free Planar Graphs

M-Degrees of Quadrangle-Free Planar Graphs M-Degrees of Quadrangle-Free Planar Graphs Oleg V. Borodin, 1 Alexandr V. Kostochka, 1,2 Naeem N. Sheikh, 2 and Gexin Yu 3 1 SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK 630090, RUSSIA E-mail: brdnoleg@math.nsc.ru

More information

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7 (67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

More information

Mining Social Network Graphs

Mining Social Network Graphs Mining Social Network Graphs Debapriyo Majumdar Data Mining Fall 2014 Indian Statistical Institute Kolkata November 13, 17, 2014 Social Network No introduc+on required Really? We s7ll need to understand

More information

CSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92.

CSE 326, Data Structures. Sample Final Exam. Problem Max Points Score 1 14 (2x7) 2 18 (3x6) 3 4 4 7 5 9 6 16 7 8 8 4 9 8 10 4 Total 92. Name: Email ID: CSE 326, Data Structures Section: Sample Final Exam Instructions: The exam is closed book, closed notes. Unless otherwise stated, N denotes the number of elements in the data structure

More information

B AB 5 C AC 3 D ABGED 9 E ABGE 7 F ABGEF 8 G ABG 6 A BEDA 3 C BC 1 D BCD 2 E BE 1 F BEF 2 G BG 1

B AB 5 C AC 3 D ABGED 9 E ABGE 7 F ABGEF 8 G ABG 6 A BEDA 3 C BC 1 D BCD 2 E BE 1 F BEF 2 G BG 1 p.9 9.5 a. Find the shortest path from A to all other vertices for the graph in Figure 9.8. b. Find the shortest unweighted path from B to all other vertices for the graph in Figure 9.8. A 5 B C 7 7 6

More information

The positive minimum degree game on sparse graphs

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

More information

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

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

More information

Kings in Tournaments. Yu Yibo, Di Junwei, Lin Min

Kings in Tournaments. Yu Yibo, Di Junwei, Lin Min Kings in Tournaments by Yu Yibo, i Junwei, Lin Min ABSTRACT. Landau, a mathematical biologist, showed in 1953 that any tournament T always contains a king. A king, however, may not exist. in the. resulting

More information

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

More information

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

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

More information

The Open University s repository of research publications and other research outputs

The Open University s repository of research publications and other research outputs Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

More information

Extremal Wiener Index of Trees with All Degrees Odd

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

More information

On the crossing number of K m,n

On the crossing number of K m,n On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

More information

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

More information

Offline sorting buffers on Line

Offline sorting buffers on Line Offline sorting buffers on Line Rohit Khandekar 1 and Vinayaka Pandit 2 1 University of Waterloo, ON, Canada. email: rkhandekar@gmail.com 2 IBM India Research Lab, New Delhi. email: pvinayak@in.ibm.com

More information

Odd induced subgraphs in graphs of maximum degree three

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

More information

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

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

More information

Max Flow, Min Cut, and Matchings (Solution)

Max Flow, Min Cut, and Matchings (Solution) Max Flow, Min Cut, and Matchings (Solution) 1. The figure below shows a flow network on which an s-t flow is shown. The capacity of each edge appears as a label next to the edge, and the numbers in boxes

More information

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8]

2. (a) Explain the strassen s matrix multiplication. (b) Write deletion algorithm, of Binary search tree. [8+8] Code No: R05220502 Set No. 1 1. (a) Describe the performance analysis in detail. (b) Show that f 1 (n)+f 2 (n) = 0(max(g 1 (n), g 2 (n)) where f 1 (n) = 0(g 1 (n)) and f 2 (n) = 0(g 2 (n)). [8+8] 2. (a)

More information

A Sublinear Bipartiteness Tester for Bounded Degree Graphs

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

More information

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

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

More information

Types of Degrees in Bipolar Fuzzy Graphs

Types of Degrees in Bipolar Fuzzy Graphs pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 4857-4866 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department

More information

Class One: Degree Sequences

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

More information

CS2 Algorithms and Data Structures Note 11. Breadth-First Search and Shortest Paths

CS2 Algorithms and Data Structures Note 11. Breadth-First Search and Shortest Paths CS2 Algorithms and Data Structures Note 11 Breadth-First Search and Shortest Paths In this last lecture of the CS2 Algorithms and Data Structures thread we will consider the problem of computing distances

More information

Social Media Mining. Graph Essentials

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

More information

arxiv:1409.4299v1 [cs.cg] 15 Sep 2014

arxiv:1409.4299v1 [cs.cg] 15 Sep 2014 Planar Embeddings with Small and Uniform Faces Giordano Da Lozzo, Vít Jelínek, Jan Kratochvíl 3, and Ignaz Rutter 3,4 arxiv:409.499v [cs.cg] 5 Sep 04 Department of Engineering, Roma Tre University, Italy

More information

Problem Set 7 Solutions

Problem Set 7 Solutions 8 8 Introduction to Algorithms May 7, 2004 Massachusetts Institute of Technology 6.046J/18.410J Professors Erik Demaine and Shafi Goldwasser Handout 25 Problem Set 7 Solutions This problem set is due in

More information

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

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

More information

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

Partitioning edge-coloured complete graphs into monochromatic cycles and paths arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

More information

Labeling outerplanar graphs with maximum degree three

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

More information

Algorithms and Data Structures (INF1) Lecture 14/15 Hua Lu

Algorithms and Data Structures (INF1) Lecture 14/15 Hua Lu Algorithms and Data Structures (INF1) Lecture 14/15 Hua Lu Department of Computer Science Aalborg University Fall 2007 This Lecture Shortest paths Problem preliminary Shortest paths in DAG Bellman-Moore

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions

Baltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.

More information

Every tree contains a large induced subgraph with all degrees odd

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

More information

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

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

More information

Monitor Placement for Maximal Identifiability in Network Tomography

Monitor Placement for Maximal Identifiability in Network Tomography Monitor Placement for Maximal Identifiability in Network Tomography Liang Ma, Ting He, Kin K. Leung, Ananthram Swami, and Don Towsley Imperial College, London, UK. Email: {l.ma10, kin.leung}@imperial.ac.uk

More information

8. Matchings and Factors

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,

More information

CMPS 102 Solutions to Homework 1

CMPS 102 Solutions to Homework 1 CMPS 0 Solutions to Homework Lindsay Brown, lbrown@soe.ucsc.edu September 9, 005 Problem..- p. 3 For inputs of size n insertion sort runs in 8n steps, while merge sort runs in 64n lg n steps. For which

More information

P. Jeyanthi and N. Angel Benseera

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

More information

CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

More information

The Max-Distance Network Creation Game on General Host Graphs

The Max-Distance Network Creation Game on General Host Graphs The Max-Distance Network Creation Game on General Host Graphs 13 Luglio 2012 Introduction Network Creation Games are games that model the formation of large-scale networks governed by autonomous agents.

More information

An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

More information

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3 Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université Paris-Dauphine, Place du Marechal de Lattre de Tassigny, F-75775 Paris

More information

Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

More information

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

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

More information

Solution Guide for Chapter 6: The Geometry of Right Triangles

Solution Guide for Chapter 6: The Geometry of Right Triangles Solution Guide for Chapter 6: The Geometry of Right Triangles 6. THE THEOREM OF PYTHAGORAS E-. Another demonstration: (a) Each triangle has area ( ). ab, so the sum of the areas of the triangles is 4 ab

More information

An Approximation Algorithm for the Unconstrained Traveling Tournament Problem

An Approximation Algorithm for the Unconstrained Traveling Tournament Problem An Approximation Algorithm for the Unconstrained Traveling Tournament Problem Shinji Imahori 1, Tomomi Matsui 2, and Ryuhei Miyashiro 3 1 Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku,

More information

Product irregularity strength of certain graphs

Product irregularity strength of certain graphs Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

More information

FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant

More information

arxiv: v2 [math.co] 30 Nov 2015

arxiv: v2 [math.co] 30 Nov 2015 PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out

More information