Katedra Algorytmów i Modelowania Systemów
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2016

An O ( n log n ) algorithm for finding edge span of cacti
Let G=(V,E) be a nonempty graph and xi be a function. In the paper we study the computational complexity of the problem of finding vertex colorings c of G such that: (1) c(u)c(v)>=xi(uv) for each edge uv of E; (2) the edge span of c, i.e. max{c(u)c(v): uv belongs to E}, is minimal. We show that the problem is NPhard for subcubic outerplanar graphs of a very simple structure (similar to cycles) and polynomially solvable for...

Applications of semidefinite optimization in quantum information protocols
This work is concerned with the issue of applications of the semidefinite programming (SDP) in the field of quantum information sci ence. Our results of the analysis of certain quantum information protocols using this optimization technique are presented, and an implementation of a relevant numerical tool is introduced. The key method used is NPA discovered by Navascues et al. [Phys. Rev. Lett. 98, 010401 (2007)]. In chapter...

Bounds on the cover time of parallel rotor walks
The rotorrouter mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, a set of k identical walkers is deployed in parallel, starting from a chosen subset of nodes, and moving around the graph in synchronous steps. During the process, each node successively propagates walkers visiting it along its outgoing arcs in roundrobin fashion, according to a fixed ordering. We consider...

Distributed Evacuation in Graphs with Multiple Exits
We consider the problem of efficient evacuation using multiple exits. We formulate this problem as a discrete problem on graphs where mobile agents located in distinct nodes of a given graph must quickly reach one of multiple possible exit nodes, while avoiding congestion and bottlenecks. Each node of the graph has the capacity of holding at most one agent at each time step. Thus, the agents must choose their movements strategy...

Edgecoloring of 3uniform hypergraphs
We consider edgecolorings of 3uniform hypergraphs which is a natural generalization of the problem of edgecolorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NPcomplete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.

Eqiuitable coloring of corona products of cubic graphs is harder than ordinary coloring
A graph is equitably kcolorable if its vertices can be partitioned into k independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G. In this paper the problem of determinig the equitable coloring number for coronas of cubic graphs is studied. Although the problem of ordinary coloring of coronas...

Equitable coloring of graphs. Recent theoretical results and new practical algorithms
In this paper we survey recent theoretical results concerning conditions for equitable colorability of some graphs and recent theoretical results concerning the complexity of equitable coloring problem. Next, since the general coloring problem is strongly NPhard, we report on practical experiments with some efficient polynomialtime algorithms for approximate equitable coloring of general graphs.

Global defensive sets in graphs
In the paper we study a new problem of finding a minimum global defensive set in a graph which is a generalization of the global alliance problem. For a given graph G and a subset S of a vertex set of G, we define for every subset X of S the predicate SEC ( X ) = true if and only if  N [ X ] ∩ S  ≥  N [ X ] \ S  holds, where N [ X ] is a closed neighbourhood of X in graph G. A set S is a defensive alliance if and only if for...

Increased Certification of Semidevice Independent Random Numbers using Many Inputs and More Postprocessing
Quantum communication with systems of dimension larger than two provides advantages in information processing tasks. Examples include higher rates of key distribution and random number generation. The main disadvantage of using such multidimensional quantum systems is the increased complexity of the experimental setup. Here, we analyze a notsoobvious problem: the relation between randomness certification and computational requirements...

Independence in uniform linear trianglefree hypergraphs
The independence number a(H) of a hypergraph H is the maximum cardinality of a set of vertices of H that does not contain an edge of H. Generalizing Shearer’s classical lower bound on the independence number of trianglefree graphs Shearer (1991), and considerably improving recent results of Li and Zang (2006) and Chishti et al. (2014), we show a new lower bound for a(H) for an runiform linear trianglefree hypergraph H with r>=2.

Lossless Compression of Binary Trees with Correlated Vertex Names
Compression schemes for advanced data structures have become the challenge of today. Information theory has traditionally dealt with conventional data such as text, image, or video. In contrast, most data available today is multitype and contextdependent. To meet this challenge, we have recently initiated a systematic study of advanced data structures such as unlabeled graphs [1]. In this paper, we continue this program by considering...

Network Graph Transformation Providing Fast Calculation of Paths for Resilient Routing
Protection of transmission against failures can be appropriately dealt with by alternative paths. However, common schemes (e.g., Bhandaris scheme) are characterized by a remarkable delay while determining the transmission paths. This in turn may have a serious impact on serving dynamic demands (characterized by relatively short duration time). As a remedy to this problem, we introduce an approach to precompute the sets of disjoint...

Nonisolating bondage in graphs
A dominating set of a graph $G = (V,E)$ is a set $D$ of vertices of $G$ such that every vertex of $V(G) \setminus D$ has a neighbor in $D$. The domination number of a graph $G$, denoted by $\gamma(G)$, is the minimum cardinality of a dominating set of $G$. The nonisolating bondage number of $G$, denoted by $b'(G)$, is the minimum cardinality among all sets of edges $E' \subseteq E$ such that $\delta(GE') \ge 1$ and $\gamma(GE')...

Normalform preemption sequences for an open problem in scheduling theory
Structural properties of optimal preemptive schedules have been studied in a number of recent papers with a primary focus on two structural parameters: the minimum number of preemptions necessary, and a tight lower bound on shifts, i.e., the sizes of intervals bounded by the times created by preemptions, job starts, or completions. These two parameters have been investigated for a large class of preemptive scheduling problems,...

On bipartization of cubic graphs by removal of an independent set
We study a new problem for cubic graphs: bipartization of a cubic graph Q by deleting sufficiently large independent set.

On some open questions for Ramsey and Folkman numbers
We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical twocolor Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5,5) and discuss the conjecture that it is equal to 43.

On the hardness of computing span of subcubic graphs
In the paper we study the problem of finding ξcolorings with minimal span, i.e. the difference between the largest and the smallest color used.

Sharp bounds for the complexity of semiequitable coloring of cubic and subcubic graphs
In this paper we consider the complexity of semiequitable kcoloring of the vertices of a cubic or subcubic graph. We show that, given nvertex subcubic graph G, a semiequitable kcoloring of G is NPhard if s >= 7n/20 and polynomially solvable if s <= 7n/21, where s is the size of maximum color class of the coloring.

Strategic balance in graphs
For a given graph G, a nonempty subset S contained in V ( G ) is an alliance iff for each vertex v ∈ S there are at least as many vertices from the closed neighbourhood of v in S as in V ( G ) − S. An alliance is global if it is also a dominating set of G. The alliance partition number of G was defined in Hedetniemi et al. (2004) to be the maximum number of sets in a partition of V ( G ) such that each set is an alliance. Similarly,...

Szeregowanie identycznych zadań na czterech procesorach jednorodnych z dwudzielnymi grafami konfliktów
Rozważono problem szeregowania n zadań jednostkowych na 4 procesorach jednorodnych o szybkościach s1>=s2>=s3>=s4. Celem szeregowania jest utworzenie najkrótszego możliwego harmonogramu. Zadania podlegają ograniczeniom zasobowym mówiącym, że niektóre pary zadań nie mogą być wykonane na tym samym procesorze. Podajemy algorytm dokładny, który rozwiązuje problem w czasie liniowym, o ile graf niezgodności jest kubiczny. Ponadto podajemy...

Topology recognition and leader election in colored networks
Topology recognition and leader election are fundamental tasks in distributed computing in networks. The first of them requires each node to find a labeled isomorphic copy of the network, while the result of the second one consists in a single node adopting the label 1 (leader), with all other nodes adopting the label 0 and learning a path to the leader. We consider both these problems in networks whose nodes are equipped with...

2015

2outerindependent domination in graphs
We initiate the study of 2outerindependent domination in graphs. A 2outerindependent dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D, and the set V(G)\D is independent. The 2outerindependent domination number of a graph G is the minimum cardinality of a 2outerindependent dominating set of G. We show that if a graph has minimum degree at least two,...

A bound on the number of middlestage crossbars in fcast rearrangeable Clos networks
In 2006 Chen and Hwang gave a necessary and sufficient condition under which a threestage Clos network is rearrangeable for broadcast connections. Assuming that only crossbars of the first stage have no fanout property, we give similar conditions for fcast Clos networks, where f is an arbitrary but fixed invariant of the network. Such assumptions are valid for some practical switching systems, e.g. highspeed crossconnects....

A TaskScheduling Approach for Efficient Sparse Symmetric MatrixVector Multiplication on a GPU
In this paper, a taskscheduling approach to efficiently calculating sparse symmetric matrixvector products and designed to run on Graphics Processing Units (GPUs) is presented. The main premise is that, for many sparse symmetric matrices occurring in common applications, it is possible to obtain significant reductions in memory usage and improvements in performance when the matrix is prepared in certain ways prior to computation....

An upper bound for the double outerindependent domination number of a tree
A vertex of a graph is said to dominate itself and all of its neighbors. A double outerindependent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V(G)\D is independent. The double outerindependent domination number of a graph G, denoted by γ_d^{oi}(G), is the minimum cardinality of a double outerindependent dominating set of G. We prove...

Bipartite theory of graphs: outerindependent domination
Let $G = (V,E)$ be a bipartite graph with partite sets $X$ and $Y$. Two vertices of $X$ are $X$adjacent if they have a common neighbor in $Y$, and they are $X$independent otherwise. A subset $D \subseteq X$ is an $X$outerindependent dominating set of $G$ if every vertex of $X \setminus D$ has an $X$neighbor in $D$, and all vertices of $X \setminus D$ are pairwise $X$independent. The $X$outerindependent domination number...

Deterministic Rendezvous in Restricted Graphs
In this paper we consider the problem of synchronous rendezvous in which two anonymous mobile entities (robots) A and B are expected to meet at the same time and point in a graph G = (V;E). Most of the work devoted to rendezvous in graphs assumes that robots have access to the same sets of nodes and edges, where the topology of connections may be initially known or unknown. In our work we assume the movement of robots is restricted...

Deviceindependent quantum key distribution based on measurement inputs
We provide an analysis of a family of deviceindependent quantum key distribution (QKD) protocols that has the following features. (a) The bits used for the secret key do not come from the results of the measurements on an entangled state but from the choices of settings. (b) Instead of a single security parameter (a violation of some Bell inequality) a set of them is used to estimate the level of trust in the secrecy of the key....

Distinguishing views in symmetric networks: A tight lower bound
The view of a node in a portlabeled network is an infinite tree encoding all walks in the network originating from this node. We prove that for any integers n ≥ D ≥ 1, there exists a portlabeled network with at most n nodes and diameter at most D which contains a pair of nodes whose (infinite) views are different, but whose views truncated to depth Omega( D log(n/ D )) are identical.

Distributed graph searching with a sense of direction
In this work we consider the edge searching problem for vertexweighted graphs with arbitrarily fast and invisible fugitive. The weight function w provides for each vertex v the minimum number of searchers required to guard v, i.e., the fugitive may not pass through v without being detected only if at least w(v) searchers are present at v. This problem is a generalization of the classical edge searching problem, in which one has...

Equitable and semiequitable coloring of cubic graphs and its application in batch scheduling
In the paper we consider the problems of equitable and semiequitable coloring of vertices of cubic graphs. We show that in contrast to the equitable coloring, which is easy, the problem of semiequitable coloring is NP complete within a broad spectrum of graph parameters. This affects the complexity of batch scheduling of unitlength jobs with cubic incompatibility graph on three uniform processors to minimize...

Fast collaborative graph exploration
We study the following scenario of online graph exploration. A team of k agents is initially located at a distinguished vertex r of an undirected graph. At every time step, each agent can traverse an edge of the graph. All vertices have unique identifiers, and upon entering a vertex, an agent obtains the list of identifiers of all its neighbors. We ask how many time steps are required to complete exploration, i.e., to make sure...

Graph security testing
Set S ⊂ V is called secure set iff ∀ X ⊂ S  N [ X ] ∩ S  ≥  N ( X ) \ S  [3]. That means that every subset of a secure set has at least as many friends (neighbour vertices in S) as enemies (neighbour vertices outside S) and will be defended in case of attack. Problem of determining if given set is secure is co −NP complete, there is no efficient algorithm solving it [3]. Property testers are algorithms that distinguish inputs...

Interval incidence graph coloring
In this paper we introduce a concept of interval incidence coloring of graphs and survey its general properties including lower and upper bounds on the number of colors. Our main focus is to determine the exact value of the interval incidence coloring number χii for selected classes of graphs, i.e. paths, cycles, stars, wheels, fans, necklaces, complete graphs and complete kpartite graphs. We also study the complexity of the...

KOALA Graph Theory Internet Service
KOALA has been created with the idea of C++ library templates, implementing a broad set of procedures in the fields of algorithmic graph theory and network problems in discreate optimization. During the C2NIWA project, a library has been greatly ectended, the code refactored and enclosed with the internet service available in the public repository of thr project. Today it contains interconnected educational materials in the form...

Minimum order of graphs with given coloring parameters
A complete kcoloring of a graph G=(V,E) is an assignment F: V > {1,...,k} of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one edge. Three extensively investigated graph invariants related to complete colorings are the minimum and maximum number of colors in a complete coloring (chromatic number χ(G) and achromatic number ψ(G), respectively),...

New potential functions for greedy independence and coloring
A potential function $f_G$ of a finite, simple and undirected graph $G=(V,E)$ is an arbitrary function $f_G : V(G) \rightarrow \mathbb{N}_0$ that assigns a nonnegative integer to every vertex of a graph $G$. In this paper we define the iterative process of computing the step potential function $q_G$ such that $q_G(v)\leq d_G(v)$ for all $v\in V(G)$. We use this function in the development of new CaroWeitype and Brookstype...

On some Zarankiewicz numbers and bipartite Ramsey Numbers for Quadrilateral
The Zarankiewicz number z ( m, n ; s, t ) is the maximum number of edges in a subgraph of K m,n that does not contain K s,t as a subgraph. The bipartite Ramsey number b ( n 1 , · · · , n k ) is the least positive integer b such that any coloring of the edges of K b,b with k colors will result in a monochromatic copy of K n i ,n i in the i th color, for some i , 1 ≤ i ≤ k . If n i = m for all i , then we denote this number by b k ( m )....

On the independence number of some strong products of cyclepowers
In the paper we give some theoretical and computational results on the third strong power of cyclepowers, for example, we have found the independence numbers alpha((C^2_10)^⊠3) = 30 and alpha((C^4 _14)^⊠3) = 14. A number of optimizations have been introduced to improve the running time of our exhaustive algorithm used to establish the independence number of the third strong power of cyclepowers. Moreover, our results establish...

On trees attaining an upper bound on the total domination number
A total dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D. The total domination number of a graph G, denoted by γ_t(G), is the minimum cardinality of a total dominating set of G. Chellali and Haynes [Total and paireddomination numbers of a tree, AKCE International Journal of Graphs and Combinatorics 1 (2004), 6975] established the following upper bound on the total domination...

On trees with equal 2domination and 2outerindependent domination numbers
For a graph G = (V,E), a subset D \subseteq V(G) is a 2dominating set if every vertex of V(G)\D$ has at least two neighbors in D, while it is a 2outerindependent dominating set if additionally the set V(G)\D is independent. The 2domination (2outerindependent domination, respectively) number of G, is the minimum cardinality of a 2dominating (2outerindependent dominating, respectively) set of G. We characterize all trees...

On trees with equal domination and total outerindependent domination numbers
For a graph G=(V,E), a subset D subseteq V(G) is a dominating set if every vertex of V(G)D has a neighbor in D, while it is a total outerindependent dominating set if every vertex of G has a neighbor in D, and the set V(G)D is independent. The domination (total outerindependent domination, respectively) number of G is the minimum cardinality of a dominating (total outerindependent dominating, respectively) set of G. We characterize...

Optimal backbone coloring of split graphs with matching backbones
For a graph G with a given subgraph H, the backbone coloring is defined as the mapping c: V(G) > N+ such that c(u)c(v) >= 2 for each edge uv \in E(H) and c(u)c(v) >= 1 for each edge uv \in E(G). The backbone chromatic number BBC(G;H) is the smallest integer k such that there exists a backbone coloring with max c(V(G)) = k. In this paper, we present the algorithm for the backbone coloring of split graphs with matching backbone.

Rendezvous of heterogeneous mobile agents in edgeweighted networks
We introduce a variant of the deterministic rendezvous problem for a pair of heterogeneous agents operating in an undirected graph, which differ in the time they require to traverse particular edges of the graph. Each agent knows the complete topology of the graph and the initial positions of both agents. The agent also knows its own traversal times for all of the edges of the graph, but is unaware of the corresponding traversal...

Robust amplification of SanthaVazirani sources with three devices
We demonstrate that amplification of arbitrarily weak randomness is possible using quantum resources. We present a randomness amplification protocol that involves Bell experiments. We find a Bell inequality which can amplify arbitrarily weak randomness and give a detailed analysis of the protocol involving it. Our analysis includes finding a sufficient violation of Bell inequality as a function of the initial quality of randomness....

The Backbone Coloring Problem for Bipartite Backbones
Let G be a simple graph, H be its spanning subgraph and λ≥2 be an integer. By a λ backbone coloring of G with backbone H we mean any function c that assigns positive integers to vertices of G in such a way that c(u)−c(v)≥1 for each edge uv∈E(G) and c(u)−c(v)≥λ for each edge uv∈E(H) . The λ backbone chromatic number BBCλ(G,H) is the smallest integer k such that there exists a λ backbone coloring c of G with backbone H satisfying...

The complexity of minimumlength path decompositions
We consider a bicriteria generalization of the pathwidth problem, where, for given integers k, l and a graph G, we ask whether there exists a path decomposition P of G such that the width of P is at most k and the number of bags in P, i.e., the length of P, is at most l. We provide a complete complexity classiﬁcation of the problem in terms of k and l for general graphs. Contrary to the original pathwidth problem, which is ﬁxedparameter...

The complexity of zerovisibility cops and robber
We consider the zerovisibility cops & robber game restricted to trees. We produce a characterisation of trees of copnumber k and We consider the computational complexity of the zerovisibility Cops and Robber game. We present a heavily modified version of an alreadyexisting algorithm that computes the zerovisibility copnumber of a tree in linear time and we show that the corresponding decision problem is NPcomplete on a nontrivial...

The computational complexity of the backbone coloring problem for boundeddegree graphs with connected backbones
Given a graph G, a spanning subgraph H of G and an integer λ>=2, a λbackbone coloring of G with backbone H is a vertex coloring of G using colors 1, 2, ..., in which the color difference between vertices adjacent in H is greater than or equal to lambda. The backbone coloring problem is to find such a coloring with maximum color that does not exceed a given limit k. In this paper, we study the backbone coloring problem for boundeddegree...

The computational complexity of the backbone coloring problem for planar graphs with connected backbones
In the paper we study the computational complexity of the backbone coloring problem for planar graphs with connected backbones. For every possible value of integer parameters λ≥2 and k≥1 we show that the following problem: Instance: A simple planar graph GG, its connected spanning subgraph (backbone) HH. Question: Is there a λbackbone coloring c of G with backbone H such that maxc(V(G))≤k? is either NPcomplete or polynomially...