For an arbitrary graph G, we consider the problem of deciding whether G is a disk graph (DG). The problem is known to be NP-hard, but it is open whether the problem actually is in NP. The problem is related to another open problem, the Polynomial Representation Hypothesis (PRH) for disk graphs: given an n-node disk graph G, can it be embedded in the plane such that the disk centers and disk rad...

We study the classical edge-searching pursuit-evasion problem where a number of pursuers have to clear a given graph of fast-moving evaders despite poor visibility, for example, where robots search a cave system to ensure that no terrorists are hiding in it. We study when polynomial-time algorithms exist to determine how many robots are needed to clear a given graph (minimum robot problem) and ...

We will give distributed approximation schemes for the maximum matching problem and the minimum connected dominating set problem in unit-disk graphs. The algorithms are deterministic, run in a poly-logarithmic number of rounds in the message passing model and the approximation error can be made O(1/ log |G|) where |G| is the order of the graph and k is a positive integer.

In the precoloring extension problem we are given a graph with some of the vertices having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-coloring of the graph. Answering an open question of Hujter and Tuza [6], we show that the precoloring extension problem is NP-complete on unit interval graphs.

Unit-Disk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each circle is specified by the coordinates of its center. Three equivalent graph models can be defined with vertices representing the circles [18]. In the intersection graph model, two vertices are adjacent if the corresponding...

We prove that every stability two unit disk graph has chromatic number at most 3 2 times its clique number.

For any graph G, the k-improper chromatic number χ(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate the ratio of the kimproper chromatic number to the clique number for unit disk graphs and random unit disk graphs to extend results of McDiarmid and Reed (1999); McDiarmid (2003) (where they considered...

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio...

An online learning problem with side information on the similarity and dissimilarity across different actions is considered. The problem is formulated as a stochastic multiarmed bandit problem with a graph-structured learning space. Each node in the graph represents an arm in the bandit problem and an edge between two nodes represents closeness in their mean rewards. It is shown that the result...

It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs (UGIGs), the recognition remains hard, as well as for all graph classes contained inbetween. The result holds even when considering only graphs with arbitrarily lar...