Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomasse make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain.Comment: 17 pages, 7 figure

Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs

We discover new hereditary classes of graphs that are minimal (with respect to set inclusion) of unbounded clique-width. The new examples include split permutation graphs and bichain graphs. Each of these classes is characterised by a finite list of minimal forbidden induced subgraphs. These, therefore, disprove a conjecture due to Daligault, Rao and Thomassé from 2010 claiming that all such minimal classes must be defined by infinitely many forbidden induced subgraphs. In the same paper, Daligault, Rao and Thomassé make another conjecture that every hereditary class of unbounded clique-width must contain a labelled infinite antichain. We show that the two example classes we consider here satisfy this conjecture. Indeed, they each contain a canonical labelled infinite antichain, which leads us to propose a stronger conjecture: that every hereditary class of graphs that is minimal of unbounded clique-width contains a canonical labelled infinite antichain

Route discovery with constant memory in oriented planar geometric networks

We address the problem of discovering routes in strongly connected planar geometric networks with directed links. Motivated by the necessity for establishing communication in wireless ad hoc networks in which the only information available to a vertex is its immediate neighborhood, we are considering routing algorithms that use the neighborhood information of a vertex for routing with constant memory only. We solve the problem for three types of directed planar geometric networks: Eulerian (in which every vertex has the same number of incoming and outgoing edges), Outerplanar (in which a single face contains all vertices of the network), and Strongly Face Connected, a new class of geometric networks that we define in the article, consisting of several faces, each face being a strongly connected outerplanar graph

On minimizing the maximum sensor movement for barrier coverage of a line segment

We consider n mobile sensors located on a line containing a barrier represented by a finite line segment. Sensors form a wireless sensor network and are able to move within the line. An intruder traversing the barrier can be detected only when it is within the sensing range of at least one sensor. The sensor network establishes barrier coverage of the segment if no intruder can penetrate the barrier from any direction in the plane without being detected. Starting from arbitrary initial positions of sensors on the line we are interested in finding final positions of sensors that establish barrier coverage and minimize the maximum distance traversed by any sensor. We distinguish several variants of the problem, based on (a) whether or not the sensors have identical ranges, (b) whether or not complete coverage is possible and (c) in the case when complete coverage is impossible, whether or not the maximal coverage is required to be contiguous. For the case of n sensors with identical range, when complete coverage is impossible, we give linear time optimal algorithms that achieve maximal coverage, both for the contiguous and non-contiguous case. When complete coverage is possible, we give an O(n 2) algorithm for an optimal solution, a linear time approximation scheme with approximation factor 2, and a (1∈+∈ε) PTAS. When the sensors have unequal ranges we show that a variation of the problem is NP-complete and identify some instances which can be solved with our algorithms for sensors with unequal ranges

Weak coverage of a rectangular barrier

Assume n wireless mobile sensors are initially dispersed in an ad hoc manner in a rectangular region. They are required to move to final locations so that they can detect any intruder crossing the region in a direction parallel to the sides of the rectangle, and thus provide weak bar-rier coverage of the region. We study three optimization problems related to the movement of sensors to achieve weak barrier coverage: minimizing the number of sensors moved (MinNum), minimizing the average distance moved by the sensors (MinSum), and minimizing the maximum distance moved by the sensors (

Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring