Cycle transversals in bounded degree graphs

Size: px
Start display at page:

Download "Cycle transversals in bounded degree graphs"

Transcription

1 Electronic Notes in Discrete Mathematics 35 (2009) 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. Protti c,d,1,3 a Universidad de Buenos Aires, FCEyN, Dpto de Computación,CONICET,Argentina b Simon Fraser University, Canada c Universidade Federal do Rio de Janeiro (UFRJ), Brazil. d Universidade Federal Fluminense (UFF), Brazil. Abstract In this work we consider the problem of finding a minimum C k -transversal (a subset of vertices hitting all the induced chordless cycles with k vertices) in a graph with bounded maximum degree. In particular, we seek for dichotomy results as follows: for a fixed value of k, finding a minimum C k -transversal is polynomial-time solvable if k p, andnp-hard otherwise. Keywords: transversal, H-transversal, H-subgraph, H-free graph 1 Introduction The graphs considered in this work are simple, connected and finite. Let H be a fixed family of graphs. An H-subgraph of a graph G is an induced subgraph of G isomorphic to a member of H. A graph is H-free if it contains no H-subgraph. An H-transversal of a graph G is a subset T V (G) such that T intersects all the H-subgraphs of G. Clearly, if T is an H-transversal of G then G T is H-free. Moreover, if T is small (minimum) then G T is a large (maximum) induced H-free subgraph of G. For a fixed family H, the general decision problem named H-transversal can be formulated as follows: given a graph G andanintegerl, decide whether G contains an H-transversal T such that T l. Yannakakis proved that this problem is NP-complete [6]. Many problems in graphs can be considered in the context of transversals. For example, if H = {C 2k+1 k 0}, then H-transversal corresponds to the maximum induced bipartite subgraph problem. The table below shows known examples. Some references for this table are [1,2,3,4,5,6]. In the table, T denotes an H-transversal of G. 1 Partially supported by Brazilian agencies CNPq and FAPERJ. 2 Partially supported by UBACyT Grants X456, Cod X143 PICT ANCyT Grant E-addresses: /$ see front matter 2009 Elsevier B.V. All rights reserved. doi: /j.endm

2 190 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) G H G T general odd cycles bipartite general {K 2} stable set general {K 3} triangle-free general {P 3} disjoint union of cliques general {P 4} cograph chordal {K 3} forest interval {K 1,3} indifference bipartite {P 4} disjoint union of bicliques chordal bipartite {C 4} forest perfect {K l } (l 1)-colorable Let k denote a fixed integer, k 3. In this work we investigate the case H = {C k }. (C k denotes a chordless cycle with k vertices.) We consider the following problem, named C k - transversal: given a graph G (with bounded maximum degree Δ) and an integer l, doesg contain a C k -transversal of size at most l? In particular, we seek for dichotomy results as follows: for a fixed value of Δ, C k -transversal is polynomial-time solvable if k p, and NP-complete otherwise. Alternatively, we can fix k and determine p such that C k -transversal is polynomial-time solvable if Δ p, and NPcomplete otherwise. The table below summarizes the complexity results dealt with in this work. Δ=2 Δ=3 Δ 4 k =3 P P NP-c k =4 P P NP-c k 5 P NP-c NP-c If Δ = 2, minimum C k -transversals are trivially obtained in polynomial time for any k, since in this case the input graph is a disjoint union of paths and cycles. In Section 2, we show that C k -transversal for maximum degree three graphs is polynomial-time solvable for k 4 and NP-complete otherwise. For maximum degree four graphs, such a dichotomy is not possible: we show in Section 3 that C k -transversal is NP-complete for any fixed k 3. This NP-completeness result trivially extends to Δ 5. In view of the hardness of finding minimum C 3 -transversals (or triangle-transversals) for Δ = 4, polynomial cases and ideas for an approximation algorithm are presented in Section 3, where we describe a decomposition theorem for maximum degree four graphs and reduction rules. 2 Maximum degree three graphs An edge e E(G) is called a k-free edge if e is contained in no induced C k of G. Theorem 2.1 C 3 -transversal is polynomial time solvable for maximum degree three graphs. Proof. Let G with Δ = 3. Clearly, if e is a 3-free edge of G then T is a triangle-transversal

3 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) of G if and only if T is a triangle-transversal of G e. Thus, to find a minimum triangletransversal of G, first remove 3-free edges; next, observe that each connected component of the remaining graph can be a triangle, a K 4 or a diamond (K 4 minus one edge). Hence a minimum triangle-transversal consists of one vertex per component. A4-bracelet is a cubic graph with vertex set V = {a 1,...,a j,b 1,...,b j } and edge set E = {a i b i 1 i j} {a i a i+1,b i b i+1 1 i j 1} {a 1 a j,b 1 b j }.Atwisted 4-bracelet is defined similarly, with edges a 1 b j,b 1 a j instead of a 1 a j,b 1 b j. Theorem 2.2 C 4 -transversal is polynomial time solvable for maximum degree three graphs. Proof. The result follows from the fact that a graph G with V (G) 6, Δ = 3 and containing no 4-free edges is a subgraph (not necessarily induced) of a bracelet or twisted bracelet. The next result completes the dichotomy for Δ = 3: Theorem 2.3 C k -transversal is NP-complete for maximum degree three graphs, for any fixed k 5. 3 Maximum degree four graphs For graphs with maximum degree four, we have: Theorem 3.1 C k -transversal is NP-complete for maximum degree four graphs, for any fixed k 3. Anaïve k-approximation algorithm is possible for finding C k -transversals in general graphs, for a fixed k 3. Given a graph G, initially set T = and C :=. At each step: (i) locate an induced C k,sayc (which can be found in polynomial time, since k is fixed); (ii) set T := T V (C) andc := C {C}; (iii) remove the vertices in V (C) fromg. Repeat (i) (iii) until there are no more C k s. Observe that the collection C is a C k -packing, thatis, a collection of vertex-disjoint C k s. Also, T is clearly a C k -transversal. If T is a minimum C k -transversal, we have T C. Since T = k C, it follows that T / T k. The above naïve algorithm produces triangle-transversals with size at most three times the optimum. Nonetheless, better behaviors might be achieved after applying some reductions on a maximum degree four input graph. We need the following definitions. A tie is a graph formed by five vertices a, b, c, d, z where d(z) =4anda, b, c, d induce 2K 2. The vertex z is called a bond. A piece is a maximum degree four connected graph containing no 3-free edges and no bonds. The following theorem characterizes pieces. Theorem 3.2 Let G be a piece. Then G is one of the graphs in Figure 1. The proof of Theorem 3.2 is a consequence of the following two lemmas. A piece G is said to be minimal if G z is not a piece for any z V (G).

4 192 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) H (n 3) n H (n 7) n G G G G G G G G G G G G Fig. 1. Pieces. The graph H n (n 3) is formed by two paths u 1 u 2...u n/2 and v 1 v 2...v n/2,plus the following edges: u i v i and u i v i+1,1 i n/2 1; u n/2 v n/2 ; and, if n is odd, u n/2 v n/2. The graph H n (n 7) is formed by a copy of H n plus the following edges: v 1 u n/2 ; v 1 v n/2 ;and u 1 v n/2 (if n is even) or u 1 v n/2 (if n is odd). Lemma 3.3 If G is a minimal piece then either G = H 3 or G = H n, for n 7. Lemma 3.4 If G is a non-minimal piece then G is one of the graphs H n (n 4),G 4, G 5i (1 i 5), G 6j (1 j 5), G 7. A direct consequence of Theorem 3.2 is: Corollary 3.5 Let G be a maximum degree four graph containing no bonds. Then a minimum triangle-transversal of G can be obtained in polynomial time. Proof. After removing the 3-free edges of G, each of its connected components is a piece, for which a minimum triangle-transversal is easily obtained. We analyze now maximum degree four graphs that may contain bonds. We can restrict our analysis to connected graphs without 3-free edges. The following definition describes a decomposition for such graphs: Definition 3.6 Let G be a maximum degree four connected graph without 3-free edges. The piece decomposition of G is the collection of pieces obtained by splitting each bond of

5 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) G into two vertices, each having two adjacent neighbors, as shown in Figure 2. Each piece of the collection is also said to be a piece of G. Fig. 2. Piece decomposition. A piece decomposition of G can be obtained in polynomial time by locating its bonds. Definition 3.7 Let G be a piece and v V (G). If d(v) = 2 then v is called a connector, otherwise an inner vertex. The template of G is a sequence (t 0,t 1,...,t k ) such that: (i) k is the number of connectors of G; (ii) if G is a piece of a graph H with minimum triangletransversal T,andi is the number of connectors of G belonging to T, then t i is the number of inner vertices of G belonging to T (observe that, for every piece, t i depends only on i, i.e., this value is independent of which group of i connectors lies in T ). For example, the templates of G 55 and G 61 are, respectively, (1, 1, 1, 0) and (2, 1, 1, 1). The template of H n,forn 4, depends on the value of n: if n =3j then it is ( n, n 3, n 3 ); if n =3j + 1 then it is ( n 1, n 1, n 4 n 2 ); and if n =3j + 2 then it is (, n 2, n 2) The piece H 3 is special, since all of its vertices are connectors and the case i = 0 cannot occur for it. To be coherent with Definition 3.7, the template of H 3 is (1, 0, 0, 0). Templates will be helpful to describe reduction rules that eliminate almost all types of pieces of a maximum degree four input graph G. Reduction rules. Let G be a maximum degree four connected graph without 3-free edges. Perform the piece decomposition of G. LetT be a minimum triangle-transversal of G to be computed, initially empty. 1. If G contains only one bond and only one piece then G is a 3-bracelet, a graph obtained from H n (for n 8) by collapsing its degree-two vertices. In this case, T is easily obtained. (In the cases below, G is not a 3-bracelet, therefore every piece of G is an induced subgraph of G.) 2. If G contains G 4 as a piece then G = G 4, since G 4 contains no connectors; hence, T can be trivially obtained. The same argument applies to G 51,G 53,G 54,G 63,G 64 and H n,for

6 194 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) n 7. Thus we can exclude these pieces from consideration. 3. For each piece H of G isomorphic to G 52, choose two inner vertices v, w V (H), one of them with degree four. Include v, w in T, and remove from G all the inner vertices of H. An analogous procedure can be applied to any piece isomorphic to G 65, provided that v, w are not adjacent to a same connector of H. 4. For each piece H of G isomorphic to G 62,letv be the connector of H and w the inner vertex of H whose neighbors induce C 4. Include v, w in T, and remove all the vertices of H from G (including v). Update the piece decomposition of G by removing v from another piece of G containing it. 5. The templates of G 7 and H 7 are identical. Thus, transform every piece H of G isomorphic to G 7 into another piece isomorphic to H 7, as follows: if v and w are the connectors of H, and xy is an edge of H such that x is adjacent to v and y is adjacent to w, then remove xy from G. 6. The template of G 61 is (2, 1, 1, 1). Note that it can be obtained by adding one to each t i in the template of H 3.Thus,ifH is a piece of G isomorphic to G 61 where v, w, x are its inner vertices, construct a graph G by removing v, w, x and adding the edges vw, vx, wx. This corresponds to replacing H by a copy of H 3. It is easy to see that there exists a triangle-transversal of G with size q if and only if there exists a triangle-transversal of G with size q +1. Thus,G 61 can be excluded from consideration. 7. For n =3j +2 (j 1), the template of H n saysthatthenumberofinnerverticesto be included in T is always the same. Thus, for each piece H isomorphic to H 3j+2 for some j 1, include in T a suitable subset of j inner vertices of H, and remove from G all the inner vertices of H. (In the case of H 5, for instance, the degree-four inner vertex must be included in T.) 8. For n =3j +1(j 2) the template of H n can be obtained by adding j 1toeacht i in thetemplateofh 4.Thus,ifH is a piece of G isomorphic to H 3j+1 for some j 2, construct agraphg by first removing all the inner vertices of H except the two neighbors v, w of some connector of H, and next linking v, w to the other connector of H. This corresponds to replacing H by a copy of H 4. Again, it is easy to see that there exists a triangle-transversal of G with size q if and only if there exists a triangle-transversal of G with size q + j 1. Thus, H 3j+1,forj 2, can also be excluded from consideration. 9. For n =3j (j 2) the template of H n can be obtained by adding j 1toeacht i in the template of H 3 (for i 2). Thus, if H is a piece of G isomorphic to H 3j for some j 2, construct a graph G by first removing all the inner vertices of H, and next creating a triangle using the connectors of H together with a new vertex x. This corresponds to replacing H by a copy of H 3. Since x is a degree-two vertex, we can assume that x/ T. Hence, there exists a triangle-transversal of G with size q if and only if there exists a triangle-transversal of G with size q + j 1. Thus, H 3j,forj 2, can also be excluded from consideration. 10. At this point, the original graph G may have been converted into a disconnected graph; then we deal with each connected component separately. Hence, let G still stand for a connected graph. The only possible pieces of G are now H 3, H 4 and G 55. In fact, we can

7 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) eliminate H 4 by adding, for each piece isomorphic to H 4 with inner vertices v and w, new vertices x, y, z and new edges xv, xw, xy, xz, yz. That is, the only pieces of G are now H 3 and G 55 (called crown). At this point, let us call G reduced graph. The application of the rules is completed. We remark that the maximum degree four graph constructed in the reduction of Theorem 3.1 contains only triangles and crowns as pieces. Hence, C 3 -transversal remains NP-complete for maximum degree four graphs containing only such pieces. By excluding the crowns, we have the following result: Theorem 3.8 C 3 -transversal is polynomial time solvable for a maximum degree four graph G when its reduced graph contains no piece isomorphic to a crown. Proof. Let G be the reduced graph of G. Construct an intersection graph P(G ) as follows: the pieces of G (triangles) are the vertices of P(G ), and two vertices of P(G )areadjacent if they share a bond of G. Take a maximum matching M of P(G ). Let S be the subset of M-unsaturated vertices of G. An optimal triangle-transversal T of G is formed as follows: for each edge e M, include in T the corresponding bond of G, and for each vertex of S include in T any vertex of the corresponding piece of G. We are currently analyzing the performance of the following approximation algorithm: given the reduced graph G, for each crown C whose degree four vertices are u C and v C, include u C in T and remove u C,v C from G. Apply to the resulting graph the method described in Theorem 3.8. References [1] D. Cornaz and A. R. Mahjoub. The maximum induced bipartite subgraph problem with edge weights. Submitted manuscript. [2] P. C. Fishburn. Interval Orders and Interval Graphs. Wiley, New York, [3] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, [4]M.C.Golumbic.Algorithmic Graph Theory and Perfect Graphs. Wiley, New York, [5] G. Manic and Y. Wakabayashi. Packing triangles in low-degree graphs and indifference graphs. Proc. European Conference on Combinatorics, Graph Theory and Applications (EuroComb 05), Berlin, Germany, Discrete Mathematics and Theoretical Computer Science (DMTCS), Vol. AE 2005, pp [6] M. Yannakakis. Node- and edge-deletion NP-complete problems. Proc. of the Tenth Annual ACM Symposium on Theory of Computing STOC 78, pp , 1978, ACM Press.

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

Clique coloring B 1 -EPG graphs

Clique coloring B 1 -EPG graphs Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La

More information

Generalized Induced Factor Problems

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.

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

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 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:

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

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

Generating models of a matched formula with a polynomial delay

Generating models of a matched formula with a polynomial delay Generating models of a matched formula with a polynomial delay Petr Savicky Institute of Computer Science, Academy of Sciences of Czech Republic, Pod Vodárenskou Věží 2, 182 07 Praha 8, Czech Republic

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

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

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

Graphs without proper subgraphs of minimum degree 3 and short cycles

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

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

Definition 11.1. Given a graph G on n vertices, we define the following quantities:

Definition 11.1. Given a graph G on n vertices, we define the following quantities: Lecture 11 The Lovász ϑ Function 11.1 Perfect graphs We begin with some background on perfect graphs. graphs. First, we define some quantities on Definition 11.1. Given a graph G on n vertices, we define

More information

Small Maximal Independent Sets and Faster Exact Graph Coloring

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

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

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

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

The Independence Number in Graphs of Maximum Degree Three

The Independence Number in Graphs of Maximum Degree Three The Independence Number in Graphs of Maximum Degree Three Jochen Harant 1 Michael A. Henning 2 Dieter Rautenbach 1 and Ingo Schiermeyer 3 1 Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684

More information

Optimal 3D Angular Resolution for Low-Degree Graphs

Optimal 3D Angular Resolution for Low-Degree Graphs Journal of Graph Algorithms and Applications http://jgaa.info/ vol. 17, no. 3, pp. 173 200 (2013) DOI: 10.7155/jgaa.00290 Optimal 3D Angular Resolution for Low-Degree Graphs David Eppstein 1 Maarten Löffler

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

Error Compensation in Leaf Power Problems

Error Compensation in Leaf Power Problems Error Compensation in Leaf Power Problems Michael Dom Jiong Guo Falk Hüffner Rolf Niedermeier April 6, 2005 Abstract The k-leaf Power recognition problem is a particular case of graph power problems: For

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

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

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,

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

M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3

M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3 Pesquisa Operacional (2014) 34(1): 117-124 2014 Brazilian Operations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/pope A NOTE ON THE NP-HARDNESS OF THE

More information

An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

More information

Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs

Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs Testing Hereditary Properties of Non-Expanding Bounded-Degree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent

More information

Finding and counting given length cycles

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

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

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

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

On the Relationship between Classes P and NP

On the Relationship between Classes P and NP Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

More information

Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

More information

THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM

THE PROBLEM WORMS (1) WORMS (2) THE PROBLEM OF WORM PROPAGATION/PREVENTION THE MINIMUM VERTEX COVER PROBLEM 1 THE PROBLEM OF WORM PROPAGATION/PREVENTION I.E. THE MINIMUM VERTEX COVER PROBLEM Prof. Tiziana Calamoneri Network Algorithms A.y. 2014/15 2 THE PROBLEM WORMS (1)! A computer worm is a standalone malware

More information

Large induced subgraphs with all degrees odd

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

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

Making life easier for firefighters

Making life easier for firefighters Making life easier for firefighters Fedor V. Fomin, Pinar Heggernes, and Erik Jan van Leeuwen Department of Informatics, University of Bergen, Norway {fedor.fomin, pinar.heggernes, e.j.van.leeuwen}@ii.uib.no

More information

NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

More information

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004 Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

More information

Removing Even Crossings

Removing Even Crossings EuroComb 2005 DMTCS proc. AE, 2005, 105 110 Removing Even Crossings Michael J. Pelsmajer 1, Marcus Schaefer 2 and Daniel Štefankovič 2 1 Department of Applied Mathematics, Illinois Institute of Technology,

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

Non-Separable Detachments of Graphs

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.

More information

Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms. Dissertation

Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms. Dissertation Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms Dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur (Dr.-Ing.) der Fakultät für Informatik und Elektrotechnik

More information

A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

More information

Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

More information

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

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

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

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

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

Minimum Bisection is NP-hard on Unit Disk Graphs

Minimum Bisection is NP-hard on Unit Disk Graphs Minimum Bisection is NP-hard on Unit Disk Graphs Josep Díaz 1 and George B. Mertzios 2 1 Departament de Llenguatges i Sistemes Informátics, Universitat Politécnica de Catalunya, Spain. 2 School of Engineering

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

5.1 Bipartite Matching

5.1 Bipartite Matching CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

More information

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

More information

Connected Identifying Codes for Sensor Network Monitoring

Connected Identifying Codes for Sensor Network Monitoring Connected Identifying Codes for Sensor Network Monitoring Niloofar Fazlollahi, David Starobinski and Ari Trachtenberg Dept. of Electrical and Computer Engineering Boston University, Boston, MA 02215 Email:

More information

Tutorial 8. NP-Complete Problems

Tutorial 8. NP-Complete Problems Tutorial 8 NP-Complete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesor-no question A decision problem

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

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs

Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Analysis of Approximation Algorithms for k-set Cover using Factor-Revealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute

More information

Resource Allocation with Time Intervals

Resource Allocation with Time Intervals Resource Allocation with Time Intervals Andreas Darmann Ulrich Pferschy Joachim Schauer Abstract We study a resource allocation problem where jobs have the following characteristics: Each job consumes

More information

P versus NP, and More

P versus NP, and More 1 P versus NP, and More Great Ideas in Theoretical Computer Science Saarland University, Summer 2014 If you have tried to solve a crossword puzzle, you know that it is much harder to solve it than to verify

More information

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs

Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Weighted Sum Coloring in Batch Scheduling of Conflicting Jobs Leah Epstein Magnús M. Halldórsson Asaf Levin Hadas Shachnai Abstract Motivated by applications in batch scheduling of jobs in manufacturing

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

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

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

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

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 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

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

The complexity of economic equilibria for house allocation markets

The complexity of economic equilibria for house allocation markets Information Processing Letters 88 (2003) 219 223 www.elsevier.com/locate/ipl The complexity of economic equilibria for house allocation markets Sándor P. Fekete a, Martin Skutella b, Gerhard J. Woeginger

More information

1 Introduction. Dr. T. Srinivas Department of Mathematics Kakatiya University Warangal 506009, AP, INDIA tsrinivasku@gmail.com

1 Introduction. Dr. T. Srinivas Department of Mathematics Kakatiya University Warangal 506009, AP, INDIA tsrinivasku@gmail.com A New Allgoriitthm for Miiniimum Costt Liinkiing M. Sreenivas Alluri Institute of Management Sciences Hanamkonda 506001, AP, INDIA allurimaster@gmail.com Dr. T. Srinivas Department of Mathematics Kakatiya

More information

Mean Ramsey-Turán numbers

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

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

Lecture 7: NP-Complete Problems

Lecture 7: NP-Complete Problems IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

More information

Heuristic Methods. Part #1. João Luiz Kohl Moreira. Observatório Nacional - MCT COAA. Observatório Nacional - MCT 1 / 14

Heuristic Methods. Part #1. João Luiz Kohl Moreira. Observatório Nacional - MCT COAA. Observatório Nacional - MCT 1 / 14 Heuristic Methods Part #1 João Luiz Kohl Moreira COAA Observatório Nacional - MCT Observatório Nacional - MCT 1 / Outline 1 Introduction Aims Course's target Adviced Bibliography 2 Problem Introduction

More information

COMS4236: Introduction to Computational Complexity. Summer 2014

COMS4236: Introduction to Computational Complexity. Summer 2014 COMS4236: Introduction to Computational Complexity Summer 2014 Mihalis Yannakakis Lecture 17 Outline conp NP conp Factoring Total NP Search Problems Class conp Definition of NP is nonsymmetric with respect

More information

Diversity Coloring for Distributed Data Storage in Networks 1

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

More information

A2 1 10-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem

A2 1 10-Approximation Algorithm for a Generalization of the Weighted Edge-Dominating Set Problem Journal of Combinatorial Optimization, 5, 317 326, 2001 c 2001 Kluwer Academic Publishers. Manufactured in The Netherlands. A2 1 -Approximation Algorithm for a Generalization of the Weighted Edge-Dominating

More information

1 Definitions. Supplementary Material for: Digraphs. Concept graphs

1 Definitions. Supplementary Material for: Digraphs. Concept graphs Supplementary Material for: van Rooij, I., Evans, P., Müller, M., Gedge, J. & Wareham, T. (2008). Identifying Sources of Intractability in Cognitive Models: An Illustration using Analogical Structure Mapping.

More information

Total colorings of planar graphs with small maximum degree

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

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

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

On parsimonious edge-colouring of graphs with maximum degree three

On parsimonious edge-colouring of graphs with maximum degree three On parsimonious edge-colouring of graphs with maximum degree three Jean-Luc Fouquet, Jean-Marie Vanherpe To cite this version: Jean-Luc Fouquet, Jean-Marie Vanherpe. On parsimonious edge-colouring of graphs

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

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

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

More information

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

More information

Exact Polynomial-time Algorithm for the Clique Problem and P = NP for Clique Problem

Exact Polynomial-time Algorithm for the Clique Problem and P = NP for Clique Problem xact Polynomial-time Algorithm for the Clique Problem and P = NP for Clique Problem Kanak Chandra Bora Department of Computer Science & ngineering Royal School of ngineering & Technology, Betkuchi, Guwahati-7810,

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

SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

More information

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) 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

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

Fairness in Routing and Load Balancing

Fairness in Routing and Load Balancing Fairness in Routing and Load Balancing Jon Kleinberg Yuval Rabani Éva Tardos Abstract We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria

More information

Graphical degree sequences and realizations

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

More information

3. Linear Programming and Polyhedral Combinatorics

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

More information

Minimum degree condition forcing complete graph immersion

Minimum degree condition forcing complete graph immersion Minimum degree condition forcing complete graph immersion Matt DeVos Department of Mathematics Simon Fraser University Burnaby, B.C. V5A 1S6 Jacob Fox Department of Mathematics MIT Cambridge, MA 02139

More information

Boulder Dash is NP hard

Boulder Dash is NP hard Boulder Dash is NP hard Marzio De Biasi marziodebiasi [at] gmail [dot] com December 2011 Version 0.01:... now the difficult part: is it NP? Abstract Boulder Dash is a videogame created by Peter Liepa and

More information

High degree graphs contain large-star factors

High degree graphs contain large-star factors High degree graphs contain large-star factors Dedicated to László Lovász, for his 60th birthday Noga Alon Nicholas Wormald Abstract We show that any finite simple graph with minimum degree d contains a

More information

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

A Graph-Theoretic Network Security Game

A Graph-Theoretic Network Security Game A Graph-Theoretic Network Security Game Marios Mavronicolas 1, Vicky Papadopoulou 1, Anna Philippou 1, and Paul Spirakis 2 1 Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus.

More information

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

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

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

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