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It is proven that the connected pathwidth of any graph G is at most 2 · pw(G) + 1, where pw(G) is the pathwidth of G. The method is constructive, i.e. it yields an efficient algorithm that for a given path decomposition of width k computes a connected path decomposition of width at most 2k + 1. The running time of the algorithm is O(dk 2), where d is the… (More)

The view of a node in a port-labeled 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 port-labeled 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 Ω(D… (More)

The rotor-router 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 maintains a cyclic ordering of its outgoing arcs, and… (More)

We consider a bi-criteria 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… (More)

We study the problem of the amount of information required to draw a complete or a partial map of a graph with unlabeled nodes and arbitrarily labeled ports. A mobile agent, starting at any node of an unknown connected graph and walking in it, has to accomplish one of the following tasks: draw a complete map of the graph, i.e., find an isomorphic copy of it… (More)

In this paper we consider the edge ranking problem of weighted trees. We prove that a special instance of this problem, namely edge ranking of multitrees is NP-hard already for multi-trees with diameter at most 10. Note that the same problem but for trees is linearly solvable. We give an O(log n)-approximation polynomial time algorithm for edge ranking of… (More)

In this paper we consider the problem of connected edge searching of weighted trees.Barrì ere et al. claim in [Capture of an intruder by mobile agents, SPAA'02 (2002) 200-209] that there exists a polynomial-time algorithm for finding an optimal search strategy, that is, a strategy that minimizes the number of used searchers. However, due to some flaws in… (More)

We consider a problem of searching an element in a partially ordered set (poset). The goal is to find a search strategy which minimizes the number of comparisons. Ben-Asher, Farchi and Newman considered a special case where the partial order has the maximum element and the Hasse diagram is a tree (tree-like posets) and they gave an O(n 4 log 3 n)-time… (More)

In this paper we consider the vertex ranking problem of weighted trees. We show that this problem is strongly NP-hard. We also give a polynomial-time reduction from the problem of vertex ranking of weighted trees to the vertex ranking of (simple) chordal graphs, which proves that the latter problem is NP-hard. In this way we solve an open problem of Aspvall… (More)

We have graph G = (V , E) and k agents initially located at node r. Agents can traverse edges of the graph according to some algorithm. Time is divided into slots. Traversal of one edge takes one time step. Agents are synchronized. Initially agents have no knowledge about the graph. Nodes of graph G have identifiers. Each agent can build a map of a part of… (More)