the second gdansk workshop on graph theorygwgt.mif.pg.gda.pl/old/pdfy/abstrakty2gwgt.pdf · the...

The Second Gdansk Workshop

on Graph Theory

List of participants and abstracts

Gdansk, June 26-28, 2014

List of participants

1. KRYSTYNA BALINSKA, e-mail: [email protected]

Technical University of Poznan (Poland)

2. ANNA BIEN, e-mail: [email protected]

University of Silesia (Poland)

3. MARTA BOROWIECKA-OLSZEWSKA, e-mail: [email protected]

University of Zielona Gora (Poland)

4. BEN-SHUNG CHOW, e-mail: [email protected]

National Sun Yet-sen University (Taiwan)

5. JANA CORONICOVA HURAJOVA, e-mail: [email protected]

Pavol Jozef Safarik University in Kosice (Slovakia)

6. JOANNA CYMAN, e-mail: [email protected]

Gdansk University of Technology (Poland)

7. KINGA DABROWSKA, e-mail: [email protected]

University of Maria Curie S lodowska in Lublin (Poland)

8. MAGDA DETTLAFF, e-mail: [email protected]


9. EWA DRGAS-BURCHARDT, e-mail: [email protected]

University of Zielona Gora (Poland)

10. TOMASZ DZIDO, e-mail: [email protected]

University of Gdansk (Poland)

11. HANNA FURMANCZYK, e-mail: [email protected]


12. ISMAEL GONZALEZ YERO, e-mail: [email protected]

University of Cadiz (Spain)

13. HARALD GROPP, e-mail: [email protected]

Heidelberg University (Germany)

2

14. HOSSEIN HAJIABOLHASSAN, e-mail: [email protected]

Technical University of Denmark (Denmark)

15. TORU HASUNUMA, e-mail: [email protected]

The University of Tokushima (Japan)

16. ELIZA JACKOWSKA, e-mail: [email protected]


17. ANDRZEJ JASTRZEBSKI, e-mail: [email protected]


18. MARCIN JURKIEWICZ, e-mail: [email protected]


19. SANDI KLAVZAR, e-mail: [email protected]

University of Ljubljana (Slovenia)

20. MARCIN KRZYWKOWSKI, e-mail: [email protected]


21. MAREK KUBALE, e-mail: [email protected]


22. MAGDALENA LEMANSKA, e-mail: [email protected]


23. ROBERT LEWON, e-mail: [email protected]


24. TOMAS MADARAS, e-mail: [email protected]


25. ANNA MA LAFIEJSKA, e-mail: [email protected]

(Poland)

26. MICHA L MA LAFIEJSKI, e-mail: [email protected]


27. MONIKA POLAK, e-mail: [email protected]


3

28. TATIANA POLLAKOVA, e-mail: [email protected]


29. KAMIL POWROZNIK, e-mail: [email protected]


30. JOANNA RACZEK, e-mail: [email protected]


31. MONIKA ROSICKA, e-mail: [email protected]


32. RINOVIA SIMANJUNTAK, e-mail: [email protected]

Bandung Institute of Technology (Indonesia)

33. MARIA JOSE SOUTO SALORIO, e-mail: [email protected]

University of A Coruna (Spain)

34. JERZY TOPP, e-mail: [email protected]


35. KRZYSZTOF TUROWSKI, e-mail: [email protected]


36. JUAN CARLOS VALENZUELA, e-mail: [email protected]

University of Cadiz (Spain)

37. TOMAS VETRIK, e-mail: [email protected]

University of the Free State (South Africa)

38. KATARZYNA WOLSKA, e-mail: [email protected]


39. RITA ZUAZUA, e-mail: [email protected]

National Autonomous University of Mexico (Mexico)

40. PAWE L ZYLINSKI, e-mail: [email protected]


4

ON THE DETERMINANT OF PLANAR GRIDS

Anna Bien

University of Silesia

e-mail: [email protected]

A singular graph is a graph whose adjacency matrix is singular i.e. its deter-minant equals zero. The problem of characterizing the structure of singular graphsoriginates from molecular chemistry. If the chemical graph related to a certainhydrocarbon is singular then the hydrocarbon is unstable.

There exists a classification of some classes of graphs. Paths Pn are singular iffn is odd, cycles Cn are singular iff 4|n.

In the talk certain methods of reduction of graphs will be discussed. The methodsare applied to calculate the determinant of hexagonal grids and cartesian productsof paths. As a results we obtain the classification of square planar grids and certainregular hexagonal grids with regard to singularity of graphs.

References

[1] L. Barrire, C. Dalf, M.A. Fiol, M. Mitjana, The generalized hierarchical productof graphs, Discrete Mathematics 309 (2009) 3871-3881

[2] A. Bien, The problem of singularity for planar grids, Discrete Mathematics 311(2011), 921-931.

[3] F. Harary, The determinant of the adjacency matrix of a graph, SIAM Rev. 4(1962), 202–210.

[4] D. Pragel, Determinants of box products of paths, Discrete Mathematics 312(2012), 1844–1847.

[5] H.M. Rara, Reduction procedures for calculating the determinant of the adja-cency matrix of some graphs and the singularity of square planar grids, DiscreteMathematics 151 (1996), 213–219.

5

ON CONSECUTIVE COLOURINGS AND DEFICIENCY OFTHE GENERALIZED HERTZ GRAPHS

Marta Borowiecka-Olszewska and Ewa Drgas-Burchardt

University of Zielona Gora

e-mail: [email protected], [email protected]

A proper edge colouring of a graph with natural numbers is consecutive if coloursof edges incident with each vertex form an interval of integers. The deficiency def(G)of a graph G is the minimum number of pendant edges whose attachment to Gmakes it consecutively colourable. In [1] Giaro, Kubale and Ma lafiejski consideredthe deficiency of the Hertz graphs. We study the deficiency of graphs from muchwider class, which we call the generalized Hertz graphs. We find the exact values ofthe deficiency of all graphs from this class. Our investigation confirms, in this class,the conjecture that the deficiency of a graph is not greater than its order. Moreover,for each generalized Hertz graph we describe necessary and sufficient conditionsthat guarantee that such a graph is consecutively colourable, and necessary andsufficient conditions that guarantee that such a graph is minimal consecutively non-colourable. Applying this result, we give the generating function for the sequencewhose specified elements represent numbers of the minimal generalized Hertz graphsthat are not consecutively colourable. In [3] one can find the sufficient condition forconsecutive non-colourability of trees in which any two leaves are in an even distance.We show that the same condition is also necessary for trees constructed on the baseof the generalized Hertz graphs.

References

[1] K. Giaro, M. Kubale, and M. Ma lafiejski, On the deficiency of bipartite graphs.Discrete Appl. Math. 94 (1999), 193–203.

[2] K. Giaro, M. Kubale, and M. Ma lafiejski, Consecutive colorings of the edges ofgeneral graphs. Discrete Math. 236 (2001), 131–143.

[3] P. A. Petrosyan and H. H. Khachatrian, Interval non-edge-colorable bipartitegraphs and multigraphs. J. Graph Theory 76 (2014), 200–216.

6

A HASSE DIAGRAM TO STUDY THE LARGESTCOLLECTION OF THE SUBSETS WITHOUT INCLUSION

Ben-shung Chow

National Sun Yet-sen University, Kaoshiung, Taiwan, ROC


There is a very large number of generalizations and analogues of the Spernertheorem [1]. Sperner theorem can be proved by LubellYamamotoMeshalkin inequal-ity [2]. We propose a similar problem replacing the conventional subsets of [n] bya permutation string. The permutation string has found many applications such asbioinformatics, stringology, and gene analysis [3]. These applications also providethe inspiration for the problems in this paper. The proposed permutation stringproblem however cannot be proved by this inequality, which would leads to a wronganswer of n factorial divided by n factorial. The reason is basically due to the differ-ence in problem setting (with the subset or with the string element). There is alsoa Hasse diagram, used to represent a finite partially ordered set, that can be asso-ciated to the proposed problem as in the Sperner theorem. It is interesting to notethat the proposed problem has the solution on the top level in its Hasse diagram,not like the generalized Sperner problems with the collection solution appearing inthe middle level.

References

[1] Jerrold R. Griggs and Linyuan Lu, On Families of Subsets With a ForbiddenSubposet, Combin., Probab. Comput. 18 (2009), 731-748.

[2] Lubell, D. (1966), ”A short proof of Sperner’s lemma”, Journal of CombinatorialTheory 1 (2) (1966), 298-299.

[3] L. Parida. Statistical significance of large gene clusters. Journal of Computa-tional Biology, 14(9) (2007),1145-1159.

7

NOTE ON THE DECAY CENTRALITY OF A GRAPH

Jana Coronicova Hurajova,

P. J. Safarik University


S. Gago and T. Madaras

Polytechnic University of Catalonia, P. J. Safarik University


The centrality indices represent a core concept for the analysis of social net-works since they help to quantify the role that a given object plays in the network.Decay centrality introduced in [1], [2] is a centrality measure based on the proxim-ity between a choosen vertex and every other vertex weighted by the decay. Moreprecisely, decay centrality of a given vertex x of a graph G is define as the sumCδ(x) =

∑y∈V (G)\{x}

δd(x,y), where d(x, y) denotes the distance between x and y and

δ ∈ (0, 1) is a parameter.We study the general properties of decay centrality, the stability of vertex ranking

depending on the choise of parameter δ and we look for the graphs whose verticesdo not change their mutual position according to this measure.

References

[1] CH. Dangalchev, Residual closeness in networks. Physica A 365 (2006), 556–564.

[2] M.O. Jackson, A. Wolinsky, A Strategic model of social and economic networks.J. of Economic Theory 71 Article no. 0108 (1996), 44–74.

8

THE RAMSEY NUMBERS FOR CYCLES VERSUSGENERALIZED FANS OR WHEELS.

Halina Bielak and Kinga Dabrowska

Institute of Mathematics, UMCS, Lublin, Poland


Recently many results for Ramsey numbers of cycles versus fans and wheels havebeen obtained. For instance Burr and Erdos [Generalization of a Ramsey-theoreticresult of Chvatal, J.Graph Theory, 7 (1983) 39–51] showed that R(C3,Wn) = 2n+ 1for n ≥ 5, Radziszowski and Xia [Paths, cycles and wheels without antitriangles,Australasian J. Comb. 9 (1994) 221–232] applied a simple method for obtaining theRamsey numbers R(C3, G), where G is either a path, a cycle or a wheel. Surahmat,Baskoro and Tomescu, [The Ramsey numbers of large cycles versus wheels, GraphsCombin. 306 (24) (2006) 3334–3337] showed that R(Cn,Wm) = 2n − 1 for even mand n ≥ 5m/2− 1 and R(Cn,Wm) = 3n− 2 for odd m and n > (5m− 9)/2.In this paper we present similar results for cycles versus some generalized fans.

9

PRIME GRAPHS

Ewa Drgas-Burchardt

University of Zielona Gora


A substitution of a graph H to a graph G instead of a vertex v ∈ V (G) is obtainedfrom disjoint G,H by first removing the vertex v from G and then making everyvertex of H adjacent with all the neighbours of v in G. An n-vertex graph is prime ifit cannot be gained by the substitution of at least 2-vertex and at most (n−1)-vertexgraph to another graph as its vertex. Let H,G1, . . . , Gn be graphs and V (H) ={v1, . . . , vn}. A graph H[G1, . . . , Gn] obtained by simultaneous substitution of Gito H instead of vi for all i ∈ [n] is called the the generalized lexicographic productof the graphs G1, . . . , Gn and the base graph H. It is a known fact that each graphis a unique composition of the generalized lexicographic products, whose all basegraphs are prime [1]. The uniqueness of this composition motivates investigation ofgraph classes that are composed only from prime graphs belonging to a fixed set.We analyse properties of prime graphs and properties of mentioned earlier graphclasses whose definition is based on the prime graph notion. A special attention ispaid to induced hereditary graph classes of this type and next to their families ofinduced forbidden graphs. Moreover, we consider the impact of small disturbancesin the definition of a graph class of this type on the family of its induced forbiddengraphs.

References

[1] T. Gallai, Transitiv orientierbare Graphen, Acta Mathematica Academiae Sci-entiarum Hungaricae 18 (1967), 25–66.

[2] E. Drgas-Burchardt, Forbidden graphs for classes of split-like graphs, EuropeanJournal of Combinatorics 39 (2014), 68–79.

[3] E. Drgas-Burchardt, On prime inductive classes of graphs, European Journalof Combinatorics 32 2 (2011), 1317–1328.

10

MONOPOLIES IN GRAPHS WITH EMPHASIS IN THEPRODUCT OF GRAPHS

Ismael Gonzalez Yero

EPS, Universidad de Cadiz, Spain


Dorota Kuziak

DEIM, Universitat Rovira i Virgili, Spain


Iztok Peterin

FEECS, University of Maribor, Slovenia


Monopolies in graphs were defined first in [1] motivated by their quite long rangeof applications in several problems related to overcoming failures. They frequentlyhave some common approaches around the notion of majorities, for instance toconsensus problems, diagnosis problems or voting systems. In this work we considersimple graphs G = (V,E). For a vertex v ∈ V and a set S ⊂ V , the notation δS(v)represents the number of neighbors v has in S and if S = V , then δV (v) is the degreeof v. The minimum degree of G is denoted by δ(G).

Given some integer k ∈{

1−⌈δ(G)2

⌉, . . . ,

⌊δ(G)2

⌋}, a vertex v of G is said to be

k-controlled by a set M if δM (v) ≥ δV (v)2 + k. Analogously, the set M is called a

k-monopoly if it k-controls every vertex v of G. Note that the case k = 0 is thestandard monopoly defined in [1].

In this article we prove that the problem of computing the minimum cardinalityof a 0-monopoly in a graph is NP-complete even restricted to bipartite or chordalgraphs. Nevertheless, for the case of trees the problem becomes polynomial. Inaddition we study the k-monopolies of some product graphs, namely, direct, strongand lexicographic product of graphs.

References

[1] N. Linial, D. Peleg, Y. Rabinovich, and M. Saks, Sphere packing and local ma-jorities in graphs, in 2nd Israel Symposium on Theory and Computing Systems(1993) 141–149.

11

CONFIGURATIONS AND GRAPHS – THE POSSIBLE POLISHCONNECTION

Harald Gropp

Heidelberg University, Germany


Configurations are linear r-regular k-uniform hypergraphs. Usually a geometriclanguage is used, the v vertices are called points or elements, the b hyperedgesare called lines or blocks. Configurations are closely related to bipartite graphsand combinatorial designs. A small interesting example is the Fano configurationor projective plane of order 2. The existence and enumeration problem will bediscussed. For which feasible parameters do configurations exist and how many arethere?

The possible Polish connection is meant twofold. There are hints that in the pastthere were Polish contributions to this research but more or less unknown. However,also for the future the participants of the Gdansk Workshop are invited to join theresearch on configurations.

12

GRAPH HOMOMORPHISMS VIA GRAPH POWERS

Hossein Hajiabolhassan and Ali Taherkhani

Technical University of Denmark


In this talk, we introduce two kinds of power for graphs [2, 2]. First, for a givengraph G, we consider G

rs , i.e., the rth power of the sth subdivision of G, and we

present some basic properties of this power. In the sequel, we introduce the graph

power G

2s+1

2r+1. We show that these powers can be considered as the dual of each

other. Precisely, we show that

G2r+12s+1 −→ H ⇐⇒ G −→ H

2s+1

2r+1.

Next, we review some coloring properties of graph powers [3].Keywords: graph homomorphism, circular coloring, fractional coloring.Subject classification: 05C

References

[1] H. Hajiabolhassan. On colorings of graph powers. Discrete Mathematics,309(13):4299–4305, 2009.

[2] Hossein Hajiabolhassan and Ali Taherkhani. Graph powers and graph homo-morphisms. Electron. J. Combin., 17(1): Research Paper 17, 2010.

[3] Hossein Hajiabolhassan and Ali Taherkhani. On the circular chromatic numberof graph powers. Journal of Graph Theory, 75 (2014), 48-58.

13

GLOBAL DEFENSIVE t-ALLIANCES IN ITERATEDSUBDIVIDED-LINE GRAPHS

Toru Hasunuma

The University of Tokushima


Let G = (V,E) be a graph that may have a self-loop. The number of edgesincident to a vertex v inG is the degree of v inG and denoted by degG(v). Let δ(G) =minv∈V (G) degG(v). A subset S ⊆ V (G) is a global defensive t-alliance in G if S is adominating set in G and for every vertex v ∈ S, |NG[v]∩S| ≥ |NG[v]−S|+ t, whereNG[v] is the closed neighborhood of v in G. We denote by γd,t(G) the cardinality ofa minimum global defensive t-alliance in G. The subdivided-line graph Γ(G) of G isdefined to be the line graph of the barycentric subdivision of G, and Γ is called thesubdivided-line graph operation [1]. The n-iterated subdivided-line graph Γn(G) is thegraph obtained from G by applying the subdivided-line graph operation n times.

We show that for 0 ≤ t ≤ δ(G) − 2, if t ≡ 1 mod 4 or t ≡ 2 mod 4 (resp.,t ≡ 0 mod 4 or t ≡ 3 mod 4) and there exists a subgraph G′ (resp., G′′) of G inwhich every even-degree vertex of G has even degree, every vertex v with degG(v) ≡1 mod 4 has odd degree (resp., no neighbor), every vertex v with degG(v) ≡ 3 mod 4has no neighbor (resp., odd degree), every odd-degree vertex of G has no self-loopwhen t is even, and every even-degree vertex of G has no self-loop when t is odd,then

γd,t(Γn(G)) =

∑v∈V (G)

⌊degG(v) + t

2

⌋degG(v)n−1

for all n ≥ 2. Our result on γd,t(Γn(G)) for t = 1 generalizes the results shown in [2]

on global strong defensive alliances in Sierpinski-like graphs S(n, k) for n ≥ 2 andS++(n, k) for n ≥ 3.

References

[1] T. Hasunuma, Structural properties of subdivided-line graphs. Proceedings of24th International Workshop on Combinatorial Algorithms (IWOCA 2013),LNCS vol. 8288 pp. 216–229, Springer-Verlag.

[2] C-H. Lin, J-J. Liu, and Y-L. Wang, Global strong defensive alliances ofSierpinski-like graphs. Theory of Comput. Syst. 53 (2013) 365–385.

14

TURAN NUMBERS OF WHEELS

Andrzej Jastrzebski

Gdansk University of Technology


The Turan number ex(n,H) is the maximum number of edges in any n-vertexgraph that does not contain graph H as a subgraph. We provide exact values ofex(n,Wk) for small n and k. We show conjectures that generalize the results.

References

[1] Bela Bollobas, Extremal graph theory. Academic Press, 1978

[2] Tomasz Dzido, A Note on Turan Numbers for Even Wheels. Graphs and Com-binatorics 29 (2013), 1305-1309.

15

DOMINATION GAME AND TOTAL DOMINATION GAME:STATE OF THE ART

Sandi Klavzar

University of Ljubljana, SloveniaUniversity of Maribor, Slovenia

Institute of Mathematics, Physics and Mechanics, Slovenia


The domination game is played on a graph G by two players, named Dominatorand Staller. They alternatively select vertices of G such that each chosen vertexenlarges the set of vertices dominated before the move on it. Dominator’s goal isthat the game is finished as soon as possible, while Staller wants the game to last aslong as possible. It is assumed that both play optimally. Game 1 and Game 2 arevariants of the game in which Dominator and Staller has the first move, respectively.The game domination number γg(G), and the Staller-start game domination numberγ′g(G), is the number of vertices chosen in Game 1 and Game 2, respectively.

This game was introduced in [1], and studied in several papers including [2, 4, 5].In this talk we will survey the knowledge we have on the game at the present. Thevery recently introduced total version of the game [3] will also be presented andcompared with the usual game.

References

[1] B. Bresar, S. Klavzar, D. F. Rall, Domination game and an imagination strat-egy, SIAM J. Discrete Math. 24 (2010) 979–991.

[2] Cs. Bujtas, Domination game on forests, arXiv:1404.1382 [math.CO], 2014.

[3] M. A. Henning, S. Klavzar, D. F. Rall, Total version of the domination game,Graphs Combin., to appear.

[4] W. B. Kinnersley, D. B. West, R. Zamani, Extremal problems for game domi-nation number, SIAM J. Discrete Math. 27 (2013) 2090–2107.

[5] G. Kosmrlj, Realizations of the game domination number, manuscript, to ap-pear in J. Comb. Optim., DOI: 10.1007/s10878-012-9572-x.

16

TREES HAVING FEW TOTAL DOMINATING SETS

Marcin Krzywkowski

Gdansk University of Technology, Polandand

University of Johannesburg, South Africa


We solve a problem posed by Professor Zdzis law Skupien at the Colourings,Independence and Domination 2013 conference.

We prove that every tree of order n ≥ 3 has at least 7√

9n total dominating sets.We also characterize all trees attaining the bound.

17

LIGHT GRAPH THEORY: A STORY

Tomas Madaras

Pavol Jozef Safarik University in Kosice, Slovakia


The study of local properties of polyhedral and plane graphs has a long andfruitful history originating from the discovery of Euler polyhedral formula in 1752and subsequent Legendre’s proof of the classical result that each polyhedral (orplane) graph contains a vertex (or a face) of degree at most 5. Various resultswhich extend this classical corollary (concerning, for example, theorems of Wernicke,Franklin, Lebesgue and Kotzig) suggest that, in the family of plane graphs, verticesand faces of small degrees tend to group in clusters. One possible way to formalizethis observation is an approach developed by light graphs theory, whose milestones,selected results and future perspectives will be presented in given talk.

18

INTERVAL INCIDENCE COLORING OF GRAPHS

Robert Janczewski, Anna Ma lafiejska and Micha l Ma lafiejski


e-mail: [email protected], [email protected], [email protected]

For a given simple graph G = (V,E), we define an incidence of G as a pair (v, e),where vertex v ∈ V is incident to edge e ∈ E. We say that two incidences (v, e),(w, f) are adjacent if one of the following holds: (1) v = w and e 6= f ; (2) e = fand v 6= w; (3) e = {v, w}, f = {w, u} and v 6= u. By an incidence coloring of G wemean a function c from the set of incidences of G to N such that c(v, e) 6= c(w, f)1 forany adjacent incidences (v, e), (w, f). Incidence coloring c is an interval incidencecoloring if and only if for every vertex v ∈ V set Ac(v) := {c(v, e) : v ∈ e ∧ e ∈ E}is an interval, i.e. contains all integers between minAc(v) and maxAc(v).

We introduce lower and upper bounds on the number of colors and determinethe exact value of the interval incidence coloring number χii for selected classes ofgraphs. We also study the complexity of the interval incidence coloring problem forsubcubic graphs for which we show that the problem of determining wheter χii ≤ 4can be solved in polynomial time, χii ≤ 5 is NP-complete, and χii ≤ 6 is trivial (i.e.interval incidence 6-coloring of subcubic graphs always exists). We also study theproblem for bipartite graphs with ∆ = 4 and we show that 5-coloring is easy and6-coloring is hard (NP-complete). Moreover, we construct an O(n∆3,5 log ∆) timeoptimal algorithm for trees.

References

[1] R.A. Brualdi, J.Q. Massey: Incidence and strong edge colorings of graphs,Discrete Math., 122 (1993), 51–58.

[2] R. Janczewski, A. Ma lafiejska, M. Ma lafiejski: Interval incidence coloring ofbipartite graphs, Discrete Appl. Math. 166 (2014), 131–140.

[3] R. Janczewski, A. Ma lafiejska, M. Ma lafiejski: Interval incidence graph coloring,Discrete Appl. Math. (2014), in press.

1To simplify notation, we write c(v, e) instead of c((v, e)) throughout the paper.

19

ON EXPANDING GRAPHS, FAMILIES OF COSPECTRALGRAPHS AND RELATED CODES

Monika Katarzyna Polak

University of Maria Curie S lodowska in Lublin


Vasyl Ustimenko

University of Maria-Curie S lodowska in Lublin


The theory of Low Density Parity Check codes started from Tanner’s observationthat bipartite graphs without cycles C4 and some other short cycles can be success-fully used for the construction of good Hamming codes. We present new examplesof LDPC codes connected with the new families of regular graphs of bounded degreeand increasing girth. Some new codes have visible advantage in comparison withcodes based on members of the family of graphs of large girth D(n, q) obtained byGuinand and Lodge. New graphs are not edge transitive. So, they are not isomor-phic to Cayley graphs or graphs from the D(n, q) family. We investigate spectralproperties of presented graphs. The experiment demonstrates existence of largespectral gaps in case of each graph. We conjecture the existence of infinite familiesof Ramanujan graphs and expanders of bounded degree, existence of strongly Ra-manujan graphs of unbounded degree. We show that new graphs can be used assource of lists of cospectral pairs of graphs of bounded or unbounded degree.

References

[1] M. Polak and V. Ustimenko, On LDPC codes based on families of expandinggraphs of increasing girth without edge-transitive automorphism groups. Com-munications in Computer and Information Science, Springer (2014), to appear.

[2] B. Bollobas, Extremal Graph Theory. Academic Press, London (1978).

[3] R. G. Gallager, Low-Density Parity-Checks Codes. Monograph, M.I.T. Press(1963).

20

SUPERMAGIC JOINS OF GRAPHS

Tatiana Pollakova

Pavol Jozef Safarik University in Kosice


A graph is called supermagic if it admits a labeling of the edges by pairwisedifferent consecutive integers such that the sum of the labels of the edges incidentwith a vertex is independent of the particular vertex. We will deal with supermagicjoins of two regular graphs.

21

STATUSES AND DOUBLE BRANCH WEIGHTSFOR SOME CLASSES OF OUTERPLANAR GRAPHS

Halina Bielak, Kamil Powroznik



We consider a connected graph G = (V (G), E(G)) with a weight function w :V (G)∪E(G)→ R+. The pair (G,w) is called a weighted graph. For any two verticesx, y in G, the weight distance between x and y, denoted by dw(x, y), is defined asdw(x, y) = min{

∑e∈E(P )w(e)}, where the minimum is taken over all paths P joining

x and y.For any vertex x ∈ V (G) we define the status of x, denoted by s(x), as follows

s(x) =∑

y∈V (G)w(y)dw(y, x). The median of G, denoted by M1(G), is the set ofvertices in G with the smallest status. The second median of G, denoted by M2(G),is the set of vertices in G with the second smallest status.

During the talk we show some metric properties of a family of weighted outer-planar graphs. For an outerplanar graph G we define some additional notions andnotations. Namely, we define the double branch weight, the first double centroid andthe second double centroid of the outerplanar graph.

We present the lower and upper bounds for statuses and double branch weightsof some outerplanar graphs. Moreover, we show some relations between medians anddouble centroids for some outerplanar graphs. In this way we extend some resultspresented in papers [1]-[3].

References

[1] Ch. Lin and J-L. Shang, Statuses and branch-weights of weighted trees. Czech.Math. J. 59 (134) (2009), 1019-1025.

[2] Ch. Lin, W-H. Tsai, J-L. Shang and Y-J. Zhang, Minimum statuses of con-nected graphs with fixed maximum degree and order. Journal of CombinatorialOptimization 24 (2012) 147-161.

[3] B. Zelinka, Medians and peripherians of trees. Archivum Mathematicum, Vol.4(1968), No.2, 87-95.

22

CONTRADICTION AND ASSIGNMENT NUMBERS

Monika Rosicka

University of Gdansk


Simone Severini

University College London


For a given graph G and labeling K assigning a permutation πe of the set[n] = {0, 1, ..., n} to each edge e of G we consider assignments k : V (G) 7→ [n].A contradiction in a vertex-assignment k is an edge uv such that πuv(k(u)) 6= k(v).We study the number of possible assignments without contradictions and the mini-mal number of contradictions over all assignments for a given G and K.

References

[1] J. Lee, M.Y. Sohn, On permutation Graphs over a graph, Comm. Korean Math.Soc. (1995), no.4, 831-837.

[2] F. Harary, On the notion of balance of a signed graph, Michigan MathematicalJournal, 2(2): 143-146, 1953.

[3] A. Agarwal, M. Charikar, K. Makarychev, Y. Makarychev, O(√

log n) approxi-mation algorithms for min UnCut, min 2CNF deletion, and directed cut problems,in: Proc 37th STOC, ACM Press, 2005, pp.573–581.

[4] Luca Trevisan, Approximation Algorithms for Unique Games, Theory of Com-puting (2008), vol. 4, pages 111-128.

23

STRONG ORIENTED GRAPHS WITHLARGEST DIRECTED METRIC DIMENSION

Yozef Tjandra and Rinovia Simanjuntak

Institut Teknologi Bandung

Let D be a strongly connected oriented graph with vertex-set V and arc-set A.The distance from a vertex u to another vertex v, d(u, v) is the minimum length oforiented paths from u to v. Suppose B = {b1, b2, b3, ...bk} is a nonempty orderedsubset of V . The representation of a vertex v with respect to B, r(v|B), is definedas a vector (d(v, b1), d(v, b2), ..., d(v, bk)). If any two distinct vertices u, v satisfyr(u|B) 6= r(v|B), then B is said to be a resolving set of D. If the cardinality of Bis minimum then B is said to be a basis of D and the cardinality of B is called thedirected metric dimension of D, dim(D).

In this talk we shall prove that ifD is a strongly connected oriented graph of ordern, then dim(D) ≤ n− 3. Furthermore, we shall characterize strong oriented graphsattaining the upper bound, i.e., oriented graphs of order n and metric dimensionn− 3.

References

[1] Melodie Fehr, Shonda Gosselin, Ortrud R. Oellermann, The metric dimensionof Cayley digraphs, Disc. Math. 306 (2006) 31-41.

[2] G. Chartrand, L. Eroh, M. Johnson, O.R. Oellermann, Resolvability in Graphsand the Metric Dimension of a Graph, Disc. Appl. Math. 105 (2000) 99-113.

[3] G. Chartrand, M. Raines, P. Zhang, The Directed Distance Dimension of Ori-ented Graphs, Math. Bohemica 125 (2000) 155-168.

[4] G. Chartrand, M. Raines, P. Zhang, On the dimension of oriented graphs, Util.Math. 60 (2001) 139-151.

[5] F. Harary and R.A. Melter, On The Metric Dimension of a Graph, Ars Combin.,2 (1976) 191-195.

[6] A. Lozano, Symmetry Breaking in Tournaments, Elec. J. Combin. 20 (2013)#P69.

[7] P.J. Slater, Leaves in Trees, Congr. Numer. 14 (1975) 549-559.

24

INDEPENDENCE NUMBER OF SOME PRODUCT GRAPHS

Marcin Jurkiewicz and Krzysztof Turowski



In the presentation, we consider the independence number of union of cycles(not only disjoint) and its products. Using some theoretical results together withoptimization algorithms we can establish some values and bounds on the previouslymentioned invariant of these product graphs.

References

[1] L. D. Baumert, R. J. McEliece, E. Rodemich, J. H. C. Rumsey, R. Stanley,and H. Taylor, A combinatorial packing problem. Computers in algebra andnumber theory, Amer. Math. Soc., SIAM-AMS Proc., Vol. IV (1971), 97–108.

[2] B. Codenotti, I. Gerace, and G. Resta, Some remarks on the Shannon capacityof odd cycles. Ars Combinatoria 66 (2003), 243–257.

25

ROMAN DOMINATION CONTRACTION NUMBER ANDSTRONG ROMAN DOMINATION IN GRAPHS

Pilar Alvarez, Ismael Gonzalez-Yero, J Carlos Valenzuela andTeresa Mediavilla

EPS de Algeciras. University of Cadiz (Spain)

e-mail: (pilar.ruiz)(ismael.gonzalez)(jcarlos.valenzuela)(teresa.mediavilla)@uca.es

Given an edge e = uv ∈ E(G) in a finite simple graph G, the graph Ge isobtained by means of contracting the edge e. More precisely, the edge e is removedand its two incident vertices, u and v, are merged into a new vertex w ∈ V (Ge),where the edges incident to w in Ge each correspond to an edge incident to eitheru or v in G. A Roman dominating function in G is a labeling of the vertices of thegraph f : V (G) → {0, 1, 2} such that any vertex v for which f(v) = 0 has at leasta neighbor w ∈ N(v) for which f(w) = 2. The minimum weight f(V ) =

∑v∈V f(v)

of a Roman dominating function f is called the Roman domination number of thegraph G, γR(G).

In this work we introduce the Roman domination contraction number, ctγR(G),which is the minimum number of edges that is necessary to contract in G in orderto decrease the Roman domination of the resulting graph. We prove general resultsand we obtain the exact value for small-diameter trees. We also slightly modify theoriginal strategy to defend the Roman Imperium [2, 3] which leads to the definitionof Roman domination in graphs [1] to derive an extension of this problem: the StrongRoman domination number, γStR(G), of a graph. We obtain some bounds for thisnew domination parameter and it is proven an existence theorem of graphs with agiven order n and a given γStR(G) = γ, when n and γ are related.

References

[1] E.J. Cokayne, P. A. Dreyer Jr., S.M. Hedetniemic, S.T. Hedetniemic:Roman domination in graphs, Discrete Math., 278 (2004), 11–22.

[2] C.S. ReVelle, K.E. Rosing: Defendens imperium romanum: a classicalproblem in military strategy, Amer. Math. Monthly, 107 (7)(2000), 585–594.

[3] I. Stewart: Defend the Roman Empire!, Sci. Amer. , 281 (6)(1999), 136–139.

26

DIRECTED CAYLEY GRAPHS OF GIVEN DEGREE ANDDIAMETER

Tomas Vetrık

University of the Free State, South Africa


Marcel Abas

Slovak University of Technology, Slovakia


A directed Cayley graph Cay(Γ, X) is specified by an underlying group Γ and bya unit-free generating set X for this group. Vertices of Cay(Γ, X) are the elementsof Γ and there is a directed edge from the vertex u to the vertex v if and only ifthere is a generator x ∈ X such that ux = v. We present the largest known directedCayley graphs of given degree and odd diameter.

27

PROPERTIES OF SOME TOPOLOGICAL INDICES ANDTRANSFORMATIONS OF GRAPHS

Halina Bielak and Katarzyna Wolska



In this paper we are going to show some properties of selected topological indices.We will present adjacent eccentric distance sum index and show the behaviour ofthat index after some transformations of graphs.

The adjacent eccentric distance sum index of the graph G is

ξsv(G) =∑

v∈V (G)

ε(v)D(v)

deg(v),

where ε(v) is the eccentricity of the vertex v, deg(v) is the degree of the vertex vand D(v) =

∑u∈V (G) d(u, v) is the sum of all distances from the vertex v.

References

[1] P. Dankelmann. W. Goddard, S. Swart, The average eccentricity of a graph andits subgraphs, Util. Math., 65 (2004) 41–51.

[2] S. Gupta, M. Singh. A. K. Madan, Application of Graph Theory: Relationsof Eccentric Connectivity Index and Wiener’s Index with AntiinflammatoryActivity, J. Math. Anal. Appl. 266 (2002) 259–268.

[3] S. Gupta, M. Singh. A. K. Madan,Eccentric distance sum: A novel graph in-variant for predicting biological and phisical properties, J. Math. Anal. Appl.275 (2002) 386–401.

[4] Hongbo Hua, Guihai Yu, Bounds for the Adjacent Eccentric Distance Sum,International Mathematical Forum, Vol. 7 (2002) no. 26, 1289–1294.

[5] A. Ilic, G. Yu, L. Feng, On eccentric distance sum of graphs, J. Math. Anal.Appl. 381 (2011) 590–600.

[6] G. Yu, L. Feng, A. Ilic, On the eccentric distance sum of trees and unicyclicgraphs, J. Math. Anal. Appl. 375 (2011) 99–107.

28

SUBDIVISION VERSUS DELETION OF EDGES IN A GRAPHAND ITS DOMINATION NUMBER

Magdalena Lemanska

Gdansk University of Technology, Poland


Joaquın Tey

Universidad Autonoma Metropolitana, Unidad Iztapalapa, Mexico


Rita Zuazua

Universidad Nacional Autonoma de Mexico, Mexico


Let e be an edge of a connected undirected graph G. As usual, we denote byγ(G) the domination number of G. The graph obtained by deleting an edge e fromG or subdividing an edge e of G is denoted by G− e or Ge, respectively. We call Ga sub-removable graph if γ(G− e) = γ(Ge) for any edge e of G. A graph G is calledan anti-sub-removable graph if γ(G− e) 6= γ(Ge) for any edge e of G.

In this work we give a construction of infinity families of sub-removable andanti-sub-removable graphs and characterize sub-removable and anti-sub-removabletrees.

29

CLEARING CONNECTIONS BY FEW AGENTS

Christos Levcopoulos and Andrzej Lingas

Lund University, S-221 00 Lund, Sweden

e-mail: {Christos.Levcopoulos,Andrzej.Lingas}@cs.lth.se

Bengt J. Nilsson

Malmo University, SE-205 06 Malmo, Sweden


Pawe l Zylinski

University of Gdansk, 80-952 Gdansk, Poland


We study the problem of clearing connections by agents placed at some verticesin a directed graph. The agents can move only along directed paths. The objectiveis to minimize the number of agents guaranteeing that any pair of vertices can beconnected by a underlying undirected path that can be cleared by the agents. Weprovide several results on the hardness, approximability and parameterized com-plexity of the problem. In particular, we show it to be: NP-hard, 2-approximablein polynomial-time, and solvable exactly in O(αn322α) time, where α is the numberof agents in the solution. In addition, we give a simple linear-time algorithm opti-mally solving the problem in digraphs whose underlying graphs are trees. Finally,we discuss a related problem, where the task is to clear with a minimum number ofagents a subgraph of the underlying graph containing its spanning tree. We showthat this problem also admits a 2-approximation in polynomial time.

30

the second gdansk workshop on graph theorygwgt.mif.pg.gda.pl/old/pdfy/abstrakty2gwgt.pdf · the...

Documents