Cycle transversals in bounded degree graphs


 Amos Blankenship
 2 years ago
 Views:
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 polynomialtime solvable if k p, andnphard otherwise. Keywords: transversal, Htransversal, Hsubgraph, Hfree graph 1 Introduction The graphs considered in this work are simple, connected and finite. Let H be a fixed family of graphs. An Hsubgraph of a graph G is an induced subgraph of G isomorphic to a member of H. A graph is Hfree if it contains no Hsubgraph. An Htransversal of a graph G is a subset T V (G) such that T intersects all the Hsubgraphs of G. Clearly, if T is an Htransversal of G then G T is Hfree. Moreover, if T is small (minimum) then G T is a large (maximum) induced Hfree subgraph of G. For a fixed family H, the general decision problem named Htransversal can be formulated as follows: given a graph G andanintegerl, decide whether G contains an Htransversal T such that T l. Yannakakis proved that this problem is NPcomplete [6]. Many problems in graphs can be considered in the context of transversals. For example, if H = {C 2k+1 k 0}, then Htransversal 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 Htransversal of G. 1 Partially supported by Brazilian agencies CNPq and FAPERJ. 2 Partially supported by UBACyT Grants X456, Cod X143 PICT ANCyT Grant Eaddresses: /$ 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} trianglefree 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 polynomialtime solvable if k p, and NPcomplete otherwise. Alternatively, we can fix k and determine p such that C k transversal is polynomialtime 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 NPc k =4 P P NPc k 5 P NPc NPc 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 polynomialtime solvable for k 4 and NPcomplete otherwise. For maximum degree four graphs, such a dichotomy is not possible: we show in Section 3 that C k transversal is NPcomplete for any fixed k 3. This NPcompleteness result trivially extends to Δ 5. In view of the hardness of finding minimum C 3 transversals (or triangletransversals) 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 kfree 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 3free edge of G then T is a triangletransversal
3 M. Groshaus et al. / Electronic Notes in Discrete Mathematics 35 (2009) of G if and only if T is a triangletransversal of G e. Thus, to find a minimum triangletransversal of G, first remove 3free 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 triangletransversal consists of one vertex per component. A4bracelet 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 4bracelet 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 4free 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 NPcomplete 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 NPcomplete for maximum degree four graphs, for any fixed k 3. Anaïve kapproximation 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 vertexdisjoint 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 triangletransversals 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 3free 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 nonminimal 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 triangletransversal of G can be obtained in polynomial time. Proof. After removing the 3free edges of G, each of its connected components is a piece, for which a minimum triangletransversal is easily obtained. We analyze now maximum degree four graphs that may contain bonds. We can restrict our analysis to connected graphs without 3free edges. The following definition describes a decomposition for such graphs: Definition 3.6 Let G be a maximum degree four connected graph without 3free 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 3free edges. Perform the piece decomposition of G. LetT be a minimum triangletransversal of G to be computed, initially empty. 1. If G contains only one bond and only one piece then G is a 3bracelet, a graph obtained from H n (for n 8) by collapsing its degreetwo vertices. In this case, T is easily obtained. (In the cases below, G is not a 3bracelet, 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 triangletransversal of G with size q if and only if there exists a triangletransversal 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 degreefour 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 triangletransversal of G with size q if and only if there exists a triangletransversal 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 degreetwo vertex, we can assume that x/ T. Hence, there exists a triangletransversal of G with size q if and only if there exists a triangletransversal 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 NPcomplete 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 Munsaturated vertices of G. An optimal triangletransversal 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 NPCompleteness. 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 lowdegree 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 edgedeletion NPcomplete problems. Proc. of the Tenth Annual ACM Symposium on Theory of Computing STOC 78, pp , 1978, ACM Press.
On the kpath cover problem for cacti
On the kpath 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 informationClique 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, FCENUBA, Buenos Aires, Argentina. b Departamento de Matemática, FCEUNLP, La
More informationGeneralized Induced Factor Problems
Egerváry Research Group on Combinatorial Optimization Technical reports TR200207. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More information8. 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 informationA 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 HaifaORANIM, Tivon 36006, Israel. AMS Subject Classification:
More informationApproximated 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 HingFung Ting 2 1 College of Mathematics and Computer Science, Hebei University,
More informationDiscrete 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 informationGenerating 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 informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More informationClass 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 informationOn 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 informationGraphs 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 informationCycles and cliqueminors in expanders
Cycles and cliqueminors 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 informationDefinition 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 informationSmall 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 informationExponential 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 klabeling of vertices of a graph G(V, E) is a function V [k].
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More information8.1 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 informationThe 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, D98684
More informationOptimal 3D Angular Resolution for LowDegree 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 LowDegree Graphs David Eppstein 1 Maarten Löffler
More informationOdd 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 longstanding
More informationError Compensation in Leaf Power Problems
Error Compensation in Leaf Power Problems Michael Dom Jiong Guo Falk Hüffner Rolf Niedermeier April 6, 2005 Abstract The kleaf Power recognition problem is a particular case of graph power problems: For
More informationCombinatorial 5/6approximation of Max Cut in graphs of maximum degree 3
Combinatorial 5/6approximation of Max Cut in graphs of maximum degree 3 Cristina Bazgan a and Zsolt Tuza b,c,d a LAMSADE, Université ParisDauphine, Place du Marechal de Lattre de Tassigny, F75775 Paris
More informationDetermination 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 informationTools for parsimonious edgecolouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR201010
Tools for parsimonious edgecolouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR201010 Tools for parsimonious edgecolouring of graphs
More informationM.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3
Pesquisa Operacional (2014) 34(1): 117124 2014 Brazilian Operations Research Society Printed version ISSN 01017438 / Online version ISSN 16785142 www.scielo.br/pope A NOTE ON THE NPHARDNESS OF THE
More informationAn 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 HongJian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,
More informationTesting Hereditary Properties of NonExpanding BoundedDegree Graphs
Testing Hereditary Properties of NonExpanding BoundedDegree Graphs Artur Czumaj Asaf Shapira Christian Sohler Abstract We study graph properties which are testable for bounded degree graphs in time independent
More informationFinding 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 informationTenacity 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 informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationMDegrees of QuadrangleFree Planar Graphs
MDegrees of QuadrangleFree 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 Email: brdnoleg@math.nsc.ru
More informationOn the Relationship between Classes P and NP
Journal of Computer Science 8 (7): 10361040, 2012 ISSN 15493636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,
More informationSplit 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 informationTHE 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 informationLarge 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 informationAn inequality for the group chromatic number of a graph
An inequality for the group chromatic number of a graph HongJian 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 informationMaking 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 informationNPCompleteness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University
NPCompleteness 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 informationJUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004
Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUSTINTIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February
More informationRemoving 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 informationThe 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 informationNonSeparable Detachments of Graphs
Egerváry Research Group on Combinatorial Optimization Technical reports TR200112. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.
More informationGraph Powers: Hardness Results, Good Characterizations and Efficient Algorithms. Dissertation
Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms Dissertation zur Erlangung des akademischen Grades DoktorIngenieur (Dr.Ing.) der Fakultät für Informatik und Elektrotechnik
More informationA 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 informationTreerepresentation of set families and applications to combinatorial decompositions
Treerepresentation of set families and applications to combinatorial decompositions BinhMinh BuiXuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no
More informationSHARP 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 informationUPPER 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 informationDiscrete 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 informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More informationMinimum Bisection is NPhard on Unit Disk Graphs
Minimum Bisection is NPhard 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 informationCycles 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 CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More information5.1 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 FordFulkerson
More informationON 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 informationConnected 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 informationTutorial 8. NPComplete Problems
Tutorial 8 NPComplete Problems Decision Problem Statement of a decision problem Part 1: instance description defining the input Part 2: question stating the actual yesorno question A decision problem
More informationEvery 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 informationAnalysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs
Analysis of Approximation Algorithms for kset Cover using FactorRevealing Linear Programs Stavros Athanassopoulos, Ioannis Caragiannis, and Christos Kaklamanis Research Academic Computer Technology Institute
More informationResource 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 informationP 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 informationWeighted 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 informationLabeling 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 informationBicolored Shortest Paths in Graphs with Applications to Network Overlay Design
Bicolored Shortest Paths in Graphs with Applications to Network Overlay Design Hongsik Choi and HyeongAh Choi Department of Electrical Engineering and Computer Science George Washington University Washington,
More informationCOUNTING 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 informationNPcomplete? NPhard? Some Foundations of Complexity. Prof. Sven Hartmann Clausthal University of Technology Department of Informatics
NPcomplete? NPhard? Some Foundations of Complexity Prof. Sven Hartmann Clausthal University of Technology Department of Informatics Tractability of Problems Some problems are undecidable: no computer
More informationShort Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles
Short Cycles make Whard problems hard: FPT algorithms for Whard Problems in Graphs with no short Cycles Venkatesh Raman and Saket Saurabh The Institute of Mathematical Sciences, Chennai 600 113. {vraman
More informationarxiv: 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 informationThe 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 information1 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 informationMean RamseyTurán numbers
Mean RamseyTurán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρmean coloring of a graph is a coloring of the edges such that the average
More informationA 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 informationLecture 7: NPComplete Problems
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NPComplete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit
More informationHeuristic 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 informationCOMS4236: 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 informationDiversity 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 informationA2 1 10Approximation Algorithm for a Generalization of the Weighted EdgeDominating 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 EdgeDominating
More information1 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 informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationA 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 sublineartime algorithm for testing whether a bounded degree graph is bipartite
More informationGRAPH 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 informationOn parsimonious edgecolouring of graphs with maximum degree three
On parsimonious edgecolouring of graphs with maximum degree three JeanLuc Fouquet, JeanMarie Vanherpe To cite this version: JeanLuc Fouquet, JeanMarie Vanherpe. On parsimonious edgecolouring of graphs
More informationOn 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 informationmost 4 Mirka Miller 1,2, Guillermo PinedaVillavicencio 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 PinedaVillavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle
More informationprinceton 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 informationExact Polynomialtime Algorithm for the Clique Problem and P = NP for Clique Problem
xact Polynomialtime 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, Guwahati7810,
More informationOn Some Vertex Degree Based Graph Invariants
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723730 ISSN 03406253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan
More informationSOLUTIONS 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 informationAll 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 informationP. 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 ONEPOINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel
More informationFairness 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 informationGraphical 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 information3. 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 informationMinimum 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 informationBoulder 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 informationHigh degree graphs contain largestar factors
High degree graphs contain largestar 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 informationF. 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 informationA GraphTheoretic Network Security Game
A GraphTheoretic 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 CY1678, Cyprus.
More informationON 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 informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationDegreeassociated reconstruction parameters of complete multipartite graphs and their complements
Degreeassociated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertexdeleted subgraphofagraphgisacard.
More information