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

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Opuscula Mathematica Vol. 32 No FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant dominating sets (liar s dominating sets) in graphs. Namely, we consider a new kind of fault a false alarm. Characterization of such fault-tolerant dominating sets in three different cases (dependent on the classification of the types of the faults) are presented. Keywords: liar s dominating set, fault-tolerant dominating set, false alarm, Hamming distance. Mathematics Subject Classification: 05C69, 94B INTRODUCTION Let G = (V, E) be a simple graph. A set D V is said to be a dominating set in G if N[v] D for every v V, where N[v] denotes the closed neighbourhood of v i.e. N[v] := {v} {u V uv E}. Fault-tolerant dominating sets (named by the author as a liar s dominating set) were introduced by P.J. Slater in [7] as follows. Consider a structure that could be represented by a graph (a computer, electrical, or sensor network, a floor-plan of a museum, a road network, etc.), where each vertex indicates some network location. In each network location there might appear some undesired event, and its location has to be determined. In some locations there are detectors (monitors, sensors) which are responsible for reporting on the presence and location of this undesired events in its closed neighbourhood. We assume that in any point of time at most one undesired event can occur. Let D V be a set of detectors. Each detector x D reports the location of an undesired event in its closed neighbourhood (if such is detected) or reports no location (if no undesired event is detected). All reports are collected in, say, the centre 751

2 752 Mateusz Nikodem of information. The final conclusion on the existence and, eventually, location of an undesired event is based on the reports of all detectors. We consider the case that under some circumstances detectors may fail in their reporting. In the basic, introduced in [7], model we assume that at most one detector x D makes a fault of type A or B, defined as follows. Definition 1.1. We distinguish the following types of detector s x D faults: Fault of type A (false negative): failing to report the existence of an undesired event in N[x]; Fault of type B (false identification): reporting a wrong location (i.e. reporting u, while an undesired event is at v u, where u, v N[x]). It is important to underline the difference between this model and the problem of fault-tolerant locating-dominating sets considered i.e. by Slater [6], where detectors are not indicating the precise location of an undesired event but just reporting that they occur in their neighbourhood. The latter one is related to locating-dominating sets [4, 5] and metric bases in graphs, considered by Harary and Melter [2]. The following definition and theorem are adapted from [3] and [7] with use of the above notation. Definition 1.2. A set D V is called a (A B)-fault-tolerant dominating set if the presence and location of an undesired event at any given vertex v V can be correctly inferred from the set of reports sent from all vertices in D, given that at most one vertex in D sends a faulty report of type A or type B. Theorem 1.3 ([3]). Set D is a (A B)-fault-tolerant dominating set if and only if the following conditions (1.1) and (1.2) are simultaneously satisfied: for every v V N[v] D 2, (1.1) for every distinct u, v V (N[u] N[v]) D 3. (1.2) Let us introduce necessary terminology of reporting vectors, to be used in further sections. Let D = {x 1,..., x m } V be a set of detectors. Then we define a reporting vector a = a(d) = (a 1,..., a m ), where a i is indicating the x i s reporting; a i V, i = 1,..., m. The notation a i = is used if x i is reporting no location of an undesired event (see Figure 1). Definition 1.4. An m-dimensional vector a u is called a u-faultless reporting vector if each of m detectors reports correctly, assuming that there is an undesired event at the vertex u. Now let us recall the Hamming distance. Definition 1.5 ([1]). Consider two vectors a, b W m where W is a given set. The function d : W m W m N such that d (a, b) := {i : a i b i } is called the Hamming distance. In our considerations W = V { }.

3 False alarms in fault-tolerant dominating sets in graphs 753 Fig. 1. The presented reporting vector is a = (v, u,, x 3, ), while a u = (u, u, u,, ) and a v = (v, v,, v, ) 2. FALSE ALARMS Let us consider a third type of fault that may occur in the report sent by a vertex in a fault-tolerant dominating set, namely a false alarm, or false positive. Definition 2.1. False alarm is a fault that occurs when a vertex x D reports the existence of an undesired event at a vertex v N[x] when, in fact, this condition does not exist at any vertex v N[x]. A false alarm is called a fault of type C. Definition 2.2. Set of detectors D V is called a k(a B C)-fault-tolerant dominating set if the presence and location of an undesired event at any given vertex v V can be correctly inferred from the set of reports sent from all vertices in D, given that at most k vertices in D sends a faulty report of type A, type B or type C. Theorem 2.3. Set D is a k(a B C)-fault-tolerant dominating set if and only if for every u V N[u] D 2k + 1. (2.1) Proof. Assume that (2.1) is satisfied. Observe that an undesired event is at u if and only if there are at least k + 1 detectors reporting u. This criterion guarantees that any location of an undesired event can be correctly identified, hence D is an k(a B C)-fault-tolerant dominating set. Conversely, if (2.1) is not satisfied, then there exists u V such that N[u] D = t 2k. Consequently, there exist t 1, t 2 {0, 1,..., k} such that t = t 1 + t 2. Then if t 1 detectors of N[u] report u and t 2 these detectors report, we cannot distinguish if there is an undesired event at u (with t 2 faults of type A) or there is no such event (with t 1 faults of type C). 3. ON SENSITIVITY AND SPECIFICITY OF DETECTORS The faults of the detectors may be caused by too low sensitivity or too low specificity. It seems natural to assume that a vertex in D has at most one of these two drawbacks. We can interpret that having too low sensitivity means the possibility of a wrong reaction if there is an undesired event in the detector s closed neighbourhood. On

4 754 Mateusz Nikodem the other hand having too low specificity can cause a wrong reaction if there is no undesired event. Assuming that the producer of detectors guarantees that at most k detectors have too low sensitivity and at most l have too low specificity, we formally define: Definition 3.1. Set of detectors D V is called a (k(a B) + lc)-fault-tolerant dominating set if the presence and location of an undesired event at any given vertex v V can be correctly inferred from the set of reports sent from all vertices in D, given that at most k vertices in D send a faulty report of type A or type B and at most l vertices in D sends a faulty report of type C. Theorem 3.2. Set D is a (k(a B) + lc)-fault-tolerant dominating set if and only if conditions (3.1) and (3.2) are simultaneously satisfied: for every v V N[v] D k + l + 1, (3.1) for every distinct u, v V, at least one of (3.2a) (3.2d) is satisfied: (3.2) (3.2a) (N[u] N[v]) D 2k + 2l + 1, (3.2b) N[u] N[v] D 2k + 1, (3.2c) N[u] D 2k + l + 1, (3.2d) N[v] D 2k + l + 1. Proof. Let D be a set satisfying conditions (3.1) and (3.2). Then the procedure of identifying the undesired events location is the following: (i) If there is no vertex reported at least l + 1 times, we conclude that there is no undesired event in G. (ii) If there is only one vertex v reported at least l + 1 times then v is the undesired event s location. (iii) If there are exactly two vertices u, v reported at least l + 1 times then one of them is the true location. The criterion of identification is following: Vertices u, v satisfying (3.2a). If d (a u, a) k + l then u is the undesired event s location, otherwise it is v. To show that, it is enough to notice that d (a u, a v ) 2k + 2l + 1, and (because at most k + l detectors can report incorrectly) d (a u, a) k + l d (a v, a) k + l + 1. Vertices u, v satisfying (3.2b). In this case we consider the reporting vector restricted to the set Y := N[u] N[v] D. (3.3) If d (a u Y, a Y ) k, then an undesired event is at u, otherwise it is at v. It is a consequence of the fact that only k detectors of Y can make a fault. Since d (a u Y, a v Y ) 2k + 1,

5 False alarms in fault-tolerant dominating sets in graphs 755 we get d (a u Y, a Y ) k d (a v Y, a Y ) k + 1. Vertices u, v satisfying (3.2c). Here we consider the reporting vector restricted to the set Moreover, let us define T = (N[u] N[v]) D. X vu := (N[v] D) \ N[u], s vu := X vu. (3.4) If s vu l then, in fact, the condition (3.2a) is satisfied, therefore we only consider the case s vu < l. The criterion is as follows: if d (a u T, a T ) k +s vu then an undesired event is at u, otherwise it is at v. Let us show correctness of this criterion. Observe that d (a u T, a v T ) 2k + l s vu. If there is an undesired event at u, then at most k detectors from N[u] D can make a fault of type A or B, and at most s vu detectors from X vu can cause a fault of type C. Then, d (a u T, a T ) k + s vu. If v is an undesired event s location then and, consequently d (a v T, a T ) k + l d (a u T, a T ) (2k + l s vu ) (k + l) = k + s vu + 1. Vertices u, v satisfying (3.2d). This case is obviously symmetric to the case (3.2c), hence the criterion of identifying the location is analogous. Namely, if d (a v T, a T ) k + s uv then the undesired event is at v, otherwise it is at u. (iv) If at least three vertices are reported more than l + 1 times, then one of them is the undesired event s location. In this case the criterion of identifying the true location is as follows: The location is the vertex w satisfying d (a w, a) k + l. We only have to show that there is exactly one vertex w with this property. Clearly, there is one vertex w such that d(a w, a) k + l because not more than k + l detectors can report incorrectly. We have to show that there is no other vertex v such that d(a v, a) k + l. Let w be the true location of an undesired event and v w be some other vertex reported at least l + 1 times. We show that d(a v, a) k + l + 1.

6 756 Mateusz Nikodem If v, w are satisfying condition (3.2a) then, due to previous arguments, d(a v, a) k + l + 1. If v, w satisfy (3.2b), then at least k +1 detectors report w (only k may report incorrectly) and at least l+1 other detectors report some vertex different than v and w. Hence d(a v, a) (k + 1) + (l + 1) > k + l + 1. If v, w satisfy (3.2c), i.e. N[v] D 2k + l + 1, then at least (2k + l + 1) (k + l) = k + 1 detectors of N[v] report the same as in a w (i.e. or w). That means that these k + 1 or more detectors report differently than in a v. Moreover, there are at least l + 1 detectors reporting some vertex different than w and v, therefore d(a v, a) (k + 1) + (l + 1) > k + l + 1. In the last case of v, w satisfying (3.2d), i.e. N[w] D 2k + l + 1, we easily observe that at least (2k + l + 1) k = k + l + 1 detectors report w, hence d(a v, a) k + l + 1. This completes the proof that set D satisfying conditions (3.1) and (3.2) is a (k(a B) + lc)-fault-tolerant dominating set. Now we show that if at least one of conditions (3.1), (3.2) is not fulfilled then D is not a (k(a B) + lc)-fault-tolerant dominating set. If (3.1) is not satisfied then for some vertex u then there exists k, l such that N[u] D = k + l, where k {0, 1,..., k}, l {0, 1,..., l}. Then clearly D is not a (k(a B) + lc)-dominating set, because if l detectors of N[u] report u and all other detectors report, we cannot state if there is an undesired event at u or not. Consider now the case when condition (3.1) is satisfied, but condition (3.2) is not (for some u, v V ). Then all the following conditions (which are negations of statements (3.2a) (3.2d)) are satisfied, s uv + s vu + Y 2k + 2l, (3.5) Y 2k, (3.6) k + l + 1 s uv + Y 2k + l, (3.7) k + l + 1 s vu + Y 2k + l, (3.8) with Y and s uv as defined in (3.3) and (3.4). First, observe that, due to (3.5),(3.7) and (3.8), s uv and s vu are not greater than l + k. Moreover, due to (3.5)-(3.8), there exists non-negative integers t u, t v such that Y = t u + t v, and t v + max{0, s uv l} k and t u + max{0, s vu l} k. Consider the following reporting: min{l, s uv } detectors of X uv and t u detectors of Y report u, min{l, s vu } detectors of X vu and another t v detectors of Y report v, all the rest, i.e. max{0, s uv l} detectors of X uv, max{0, s vu l} detectors of X vu and all the other detectors report,

7 False alarms in fault-tolerant dominating sets in graphs 757 Fig. 2. The case when s uv > l and s vu l. Both u and v are possible location of an undesired event In this case both u and v are a possible undesired event s location (Figure 2), namely: an event is at u, while max{0, s uv l} detectors of X uv make a fault of type A, t v detectors of Y make a fault of type B and min{l, s vu } detectors of X vu make a fault of type C, or an event is at v, while max{0, s vu l} detectors of X vu make a fault of type A, t u detectors of Y make a fault of type B and min{l, s uv } detectors of X uv make a fault of type C. This shows that if conditions (3.1) and (3.2) are not simultaneously satisfied then D is not (k(a B) + lc)-fault-tolerant dominating set. Now let us present some modifications to the interpretation of detectors faults. Namely, one can assume that having too low sensitivity means the possibility of reporting no undesired event, if there is one in a closed neighbourhood (fault of type A). On the other hand having too low precision (or specificity) means the possibility of reporting the wrong location (faults of type B or C). Definition 3.3. Set of detectors D V is called a (ka + l(b C))-fault-tolerant dominating set if the presence and location of an undesired event at any given vertex v V can be correctly inferred from the set of reports sent from all vertices in D, given that at most k vertices in D sends a faulty report of type A and at most l vertices in D sends a faulty report of type B or type C.

8 758 Mateusz Nikodem Theorem 3.4. Set D is a (ka + l(b C))-fault-tolerant dominating set if and only if conditions (3.9) and (3.10) are simultaneously satisfied: for every v V N[v] D k + l + 1, (3.9) for every distinct u, v V at least one of (3.10a) (3.10c) is satisfied: (3.10) (3.10a) (N[u] N[v]) D 2k + 2l + 1, (3.10b) N[u] D k + 2l + 1, (3.10c) N[v] D k + 2l + 1. Proof. Let D be a set of vertices for which conditions (3.9) and (3.10) are satisfied. Then the procedure of identifying an undesired event s location is as follows: (i) If not more than l detectors report the existence of an undesired event, then there is no such event in G. Otherwise there is an event somewhere in G, which is a simple consequence of the assumption that not more than l faults of type C can occur. Now consider only the case of the existence of an undesired event in G. (ii) If there is some vertex v reported at least l + 1 times then an undesired event is clearly at v. It is easy to observe that there is no possibility that more than one vertex is reported more than l times. (iii) If there is no vertex reported more than l times then an undesired event is in only one vertex u such that d(a, a u ) k + l. We have to show that there exists exactly one vertex u with this property. To the contrary assume that there exists w u such that d(a, a w ) k + l. If u, w are satisfying (3.10a) then d(a u, a w ) 2k + 2l + 1 what leads us to a contradiction to our assumptions that d(a, a u ) k + l and d(a, a w ) k + l. If u, w are satisfying (3.10b), i.e. N[u] D k + 2l + 1, then d(a, a u ) k + l implies that u is reported at least l + 1 times - a contradiction. Analogously we obtain a contradiction if u, w are satisfying (3.10c). This shows that a set fulfilling the conditions (3.9) and (3.10) is indeed a (ka + l(b C))-fault-tolerant dominating set. Now let us show the necessity of conditions (3.9) and (3.10). If D is not satisfying (3.9) then some vertex u has not more than k + l detectors in its closed neighbourhood, where k, l are such non-negative integers that k k and l l. Then, if l detectors from N[u] report u and all the rest report we cannot state if there is an undesired event at u or not. Assume now that (3.9) is satisfied and (3.10) is not. It means that there exist u, v V such that: s uv + s vu + Y 2k + 2l, (3.11) k + l + 1 s uv + Y k + 2l, (3.12) k + l + 1 s vu + Y k + 2l, (3.13) where Y and s uv are defined in (3.3) and (3.4). For clarity we analyse two subcases.

9 False alarms in fault-tolerant dominating sets in graphs 759 Subcase 1. t 2l. Consider the following reporting: l detectors from Y report u, another l detectors from Y reports v, the rest of the detectors reports. Due to (3.12) and (3.13) we notice that not more than k detectors from N[u] D and no more than k detectors from N[v] D report. Thus an undesired event might be at u or might be at v in both possibilities there are not more than k faults of type A and l faults of type B or C. Subcase 2. t < 2l. Observe that due to (3.11)-(3.13) there exist non-negative integers t u, t v such that t u + t v = t, where t u + max{0, s uv k} l and t v + max{0, s vu k} l. Consider the following reporting (Figure 3): t u detectors of Y and max{0, s uv k} of X uv report u, another t v detectors of Y and max{0, s vu k} of X vu report v, min{k, s uv } of X uv, min{k, s vu } of X vu and all the rest report. In that case we also cannot be sure if there is an undesired event at u or it is at v in both possibilities there are not more than k faults of type A and not more than l faults of type B or C, which ends the proof. Fig. 3. The case when s uv > k and s vu k. Both u and v are possible locations of an undesired event Acknowledgments The research was partially supported by the Polish Ministry of Science and Higher Education.

10 760 Mateusz Nikodem REFERENCES [1] J. Daintith, E. Wright, eds, A Dictionary of Computing, Oxford University Press, USA; 6th ed., 2010, p. 87. [2] F. Harary, R.A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), [3] M.L. Roden, P.J. Slater, Liar s domination in graphs, Discrete Math. 309 (2009) 19, [4] P.J. Slater, Domination and location in acyclic graphs, Networks 17 (1987), [5] P.J. Slater, Domination and reference sets in graphs, J. Math. Phys. Sci. 22 (1988), [6] P.J. Slater, Fault-tolerant locating-dominating sets, Discrete Math. 249 (2002), [7] P.J. Slater, Liar s domination, Networks 54 (2009) 2, Mateusz Nikodem AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30, Krakow, Poland Received: June 8, Revised: May 15, Accepted: May 16, 2012.

### 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

### 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

### 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

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

### 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

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

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

### 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

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

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

### Types of Degrees in Bipolar Fuzzy Graphs

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

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

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

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

### 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

### A simple criterion on degree sequences of graphs

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

### 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

### 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.,

### 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

### 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

### 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

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### 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

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

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

### ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION. E. I. Pancheva, A. Gacovska-Barandovska

Pliska Stud. Math. Bulgar. 22 (2015), STUDIA MATHEMATICA BULGARICA ON LIMIT LAWS FOR CENTRAL ORDER STATISTICS UNDER POWER NORMALIZATION E. I. Pancheva, A. Gacovska-Barandovska Smirnov (1949) derived four

### Product irregularity strength of certain graphs

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

### 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

### Competition Parameters of a Graph

AKCE J. Graphs. Combin., 4, No. 2 (2007), pp. 183-190 Competition Parameters of a Graph Suk J. Seo Computer Science Department Middle Tennessee State University Murfreesboro, TN 37132, U.S.A E-mail: sseo@mtsu.edu

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### 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

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### 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,

### Special Classes of Divisor Cordial Graphs

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

### Online Appendix to. Stability and Competitive Equilibrium in Trading Networks

Online Appendix to Stability and Competitive Equilibrium in Trading Networks John William Hatfield Scott Duke Kominers Alexandru Nichifor Michael Ostrovsky Alexander Westkamp Abstract In this appendix,

### 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

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

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

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

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

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

### 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

### ADDITIVE GROUPS OF RINGS WITH IDENTITY

ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsion-free

### Minimum-Density Identifying Codes in Square Grids

Minimum-Density Identifying Codes in Square Grids Marwane Bouznif, Frédéric Havet, Myriam Preissmann To cite this version: Marwane Bouznif, Frédéric Havet, Myriam Preissmann. Minimum-Density Identifying

### No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

### 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

### Bounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances

Bounded Cost Algorithms for Multivalued Consensus Using Binary Consensus Instances Jialin Zhang Tsinghua University zhanggl02@mails.tsinghua.edu.cn Wei Chen Microsoft Research Asia weic@microsoft.com Abstract

### Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

Wan Boundary Value Problems (2015) 2015:239 DOI 10.1186/s13661-015-0508-0 R E S E A R C H Open Access Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator

### The Role of Dispute Settlement Procedures in International Trade Agreements: Online Appendix

The Role of Dispute Settlement Procedures in International Trade Agreements: Online Appendix Giovanni Maggi Yale University, NBER and CEPR Robert W. Staiger Stanford University and NBER November 2010 1.

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Single machine parallel batch scheduling with unbounded capacity

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

### 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

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### The degree, size and chromatic index of a uniform hypergraph

The degree, size and chromatic index of a uniform hypergraph Noga Alon Jeong Han Kim Abstract Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the

### 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

### CHAPTER 2. Inequalities

CHAPTER 2 Inequalities In this section we add the axioms describe the behavior of inequalities (the order axioms) to the list of axioms begun in Chapter 1. A thorough mastery of this section is essential

### Degree Hypergroupoids Associated with Hypergraphs

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

### 2.3 Scheduling jobs on identical parallel machines

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

### Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

### Fuzzy Differential Systems and the New Concept of Stability

Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

### 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

### 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

### 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

### Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients

DOI: 10.2478/auom-2014-0007 An. Şt. Univ. Ovidius Constanţa Vol. 221),2014, 73 84 Irreducibility criteria for compositions and multiplicative convolutions of polynomials with integer coefficients Anca

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### A Hajós type result on factoring finite abelian groups by subsets II

Comment.Math.Univ.Carolin. 51,1(2010) 1 8 1 A Hajós type result on factoring finite abelian groups by subsets II Keresztély Corrádi, Sándor Szabó Abstract. It is proved that if a finite abelian group is

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

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

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

### Graph Security Testing

JOURNAL OF APPLIED COMPUTER SCIENCE Vol. 23 No. 1 (2015), pp. 29-45 Graph Security Testing Tomasz Gieniusz 1, Robert Lewoń 1, Michał Małafiejski 1 1 Gdańsk University of Technology, Poland Department of

### 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

### Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

### 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,

### Analysis of Algorithms, I

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

### 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

### Interpreting Kullback-Leibler Divergence with the Neyman-Pearson Lemma

Interpreting Kullback-Leibler Divergence with the Neyman-Pearson Lemma Shinto Eguchi a, and John Copas b a Institute of Statistical Mathematics and Graduate University of Advanced Studies, Minami-azabu

### Decision-making with the AHP: Why is the principal eigenvector necessary

European Journal of Operational Research 145 (2003) 85 91 Decision Aiding Decision-making with the AHP: Why is the principal eigenvector necessary Thomas L. Saaty * University of Pittsburgh, Pittsburgh,

### Removing Partial Inconsistency in Valuation- Based Systems*

Removing Partial Inconsistency in Valuation- Based Systems* Luis M. de Campos and Serafín Moral Departamento de Ciencias de la Computación e I.A., Universidad de Granada, 18071 Granada, Spain This paper

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

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

### On the representability of the bi-uniform matroid

On the representability of the bi-uniform matroid Simeon Ball, Carles Padró, Zsuzsa Weiner and Chaoping Xing August 3, 2012 Abstract Every bi-uniform matroid is representable over all sufficiently large

### 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

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Two-Sided Matching Theory

Two-Sided Matching Theory M. Utku Ünver Boston College Introduction to the Theory of Two-Sided Matching To see which results are robust, we will look at some increasingly general models. Even before we

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

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

### SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

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

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### 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

### LOOKING FOR A GOOD TIME TO BET

LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any time-but once only- you may bet that the next card

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### 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

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

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

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

### Strong Ramsey Games: Drawing on an infinite board

Strong Ramsey Games: Drawing on an infinite board Dan Hefetz Christopher Kusch Lothar Narins Alexey Pokrovskiy Clément Requilé Amir Sarid arxiv:1605.05443v2 [math.co] 25 May 2016 May 26, 2016 Abstract

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Examination paper for MA0301 Elementær diskret matematikk

Department of Mathematical Sciences Examination paper for MA0301 Elementær diskret matematikk Academic contact during examination: Iris Marjan Smit a, Sverre Olaf Smalø b Phone: a 9285 0781, b 7359 1750

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Factoring Patterns in the Gaussian Plane

Factoring Patterns in the Gaussian Plane Steve Phelps Introduction This paper describes discoveries made at the Park City Mathematics Institute, 00, as well as some proofs. Before the summer I understood

### INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem