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The maximum independent set problem is NP-complete for graphs in general, but becomes solvable in polynomial time when restricted to graphs in many special classes. The problem is also intractable from a parameterized point of view. However, very little is known about parameterized complexity of the problem in restricted graph classes. In the present paper, we analyse two techniques that have previously...
A run is an inclusion maximal occurrence in a string (as a subinterval) of a repetition v with a period p such that 2p ≤ |v|. The exponent of a run is defined as |v|/p and is ≥ 2. We show new bounds on the maximal sum of exponents of runs in a string of length n. Our upper bound of 4.1 n is better than the best previously known proven bound of 5.6 n by Crochemore & Ilie (2008). The lower bound...
A path-based support of a hypergraph H is a graph with the same vertex set as H in which each hyperedge induces a Hamiltonian subgraph. While it is -complete to compute a path-based support with the minimum number of edges or to decide whether there is a planar path-based support, we show that a path-based tree support can be computed in polynomial time if it exists.
L(2,1)-labeling is graph labeling model where adjacent vertices get labels that differ by at least 2 and vertices in distance 2 get different labels. In this paper we present an algorithm for finding an optimal L(2,1)-labeling (i.e. an L(2,1)-labeling in which largest label is the least possible) of a graph with time complexity O* ( 3.5616 n), which improves...
Introduction The introduction of tree-width by Robertson and Seymour [7] was a breakthrough in the design of graph algorithms. A lot of research since then has focused on obtaining a width measure which would be more general and still allowed efficient algorithms for a wide range of NP-hard problems on graphs of bounded width. To this end, Oum and Seymour have proposed rank-width, which allows the...
Partial words are sequences over a finite alphabet that may contain some undefined positions called holes. In this paper, we consider unavoidable sets of partial words of equal length. We compute the minimum number of holes in sets of size three over a binary alphabet (summed over all partial words in the sets). We also construct all sets that achieve this minimum. This is a step towards the difficult...
We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships...
In this paper, we consider the unordered pseudo-tree matching problem, which is a problem of, given two unordered labeled trees P and T, finding all occurrences of P in T via such many-one embeddings that preserve node labels and parent-child relationship. This problem is closely related to tree pattern matching problem for XPath queries with child axis only. If m > w , we present an efficient...
We study a (k + 1)-coloring problem in a class of (k,s)-dart graphs, k,s ≥ 2, where each vertex of degree at least k + 2 belongs to a (k,i)-diamond, i ≤ s. We prove that dichotomy holds, that means the problem is either NP-complete (if k < s), or can be solved in linear time (if k ≥ s). In particular, in the latter case we generalize the classical Brooks Theorem, that means we prove that a (k,...
In this paper, we explore worst-case solutions for the problems of pattern and multi-pattern matching on strings in the RAM model with word length w. In the first problem, we have a pattern p of length m over an alphabet of size σ, and given any text T of length n, where each character is encoded using logσ bit, we wish to find all occurrences of p. For the multi-pattern matching problem we have a...
A (2,1)-total labeling of a graph G is an assignment f from the vertex set V(G) and the edge set E(G) to the set {0,1,...,k} of nonnegative integers such that |f(x) − f(y)| ≥ 2 if x is a vertex and y is an edge incident to x, and |f(x) − f(y)| ≥ 1 if x and y are a pair of adjacent vertices or a pair of adjacent edges, for all x and y in V(G) ∪ E(G). The (2,1)-total labeling number $\lambda^T_2(G)$...
Modern computers have several levels of memory hierarchy. To obtain good performance on these processors it is necessary to design algorithms that minimize I/O traffic to slower memories in the hierarchy. In this paper, we present I/O efficient algorithms to pebble r-pyramids and derive lower bounds on the number of I/O steps to do so. The r-pyramid graph models financial applications which are of...
We know that k -Uniform Nash is W[2]-Complete when we consider imitation symmetric win-lose games (with k as the parameter) even when we have two players. However, this paper provides positive results regarding Nash equilibria. We show that consideration of sparse games or limitations of the support result in fixed-parameter algorithms with respect to one parameter only for the k -Uniform Nash problem...
Given an undirected graph G = (V,E) with a capacity function $w : E \longrightarrow \mathbb{Z}^+$ on the edges, the sparsest cut problem is to find a vertex subset S ⊂ V minimizing ∑ e ∈ E(S,V ∖ S)w(e)/(|S||V ∖ S|). This problem is NP-hard. The proof can be found in [16]. In the case of unit capacities (i. e. if w(e) = 1 for every e ∈ E) the problem is to minimize |E(...
We study the approximation complexity of the Metric Dimension problem in bounded degree, dense as well as in general graphs. For the general case, we prove that the Metric Dimension problem is not approximable within $(\!1\!-\!\epsilon\!)\!\ln\!n$ for any , unless $NP\!\subseteq\!DTIME(\!n^{\log\!\log\!n}\!)$ , and we give an approximation algorithm which matches the...
When the environment does not allow to access directly to disseminated data, a sensor network could be one of the most appropriate solution to retrieve the map of interesting areas. Based on existing approaches, we start our study from the standard random deployment of a sensor network and then we consider a coarse-grain localization algorithm which associates sensors with coordinates related to a...
Given an undirected graph with weights on its vertices, the k most vital nodes independent set problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets. We also consider the complementary problem, minimum node blocker independent set that consists of removing a subset of vertices of minimum size such that the maximum...
A homomorphism from a graph G to a graph R is locally surjective if its restriction to the neighborhood of each vertex of G is surjective. Such a homomorphism is also called an R-role assignment of G. Role assignments have applications in distributed computing, social network theory, and topological graph theory. The Role Assignment problem has as input a pair of graphs (G,R) and asks whether G has...
Given a connected graph G and a set F of faulty vertices of G, let G − F be the graph obtained from G by deletion of all vertices of F and edges incident with them. Is there an algorithm, whose running time may be bounded by a polynomial function of |F| and log|V(G)|, which decides whether G − F is still connected? Even though the answer to this question is negative in general, we describe an algorithm...
Recently we have developed a method excluding certain subgraphs from a smallest counterexample to the 5-flow conjecture. This is based on comparing ranks of two matrices of large size. The aim of this paper is to be more effective by applying these methods so that we reduce the size of matrices used in the computation.
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