In this note, a constructive proof that the classes of proper interval graphs and unit interval graphs coincide is given, a result originally established by Fred S. Roberts. Additionally, the proof yields a linear-time and space algorithm to compute a unit interval representation, given a proper interval graph as input. © 2007 Elsevier B.V. All rights reserved.

In this work we present three new recognition algorithms for circular-arc graphs and for two subclasses of this graph class. We give a linear-time recognition algorithm for circular-arc graphs based on the algorithm of Eschen and Spinrad [ES93, Esc97]. Our algorithm is simpler than the earlier linear-time recognition algorithm of McConnell [McC03], which is the only linear time recognition algo...

We study a variant of intersection representations with unit balls, that is, unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far sets, the goal is to represent the vertices of the graph by unit balls so that the balls representing two adjacent vertices intersect if and only if the corresponding edge is near. ...

We consider a well-studied generalization of the maximum clique problem which is defined as follows. Given graph G on n vertices and fixed parameter $$d\ge 1$$ , in diameter-bounded subgraph (MaxDBS for short) goal to find (vertex) diameter at most d. For $$d=1$$ this equivalent thus it NP-hard approximate within factor $$n^{1-\epsilon }$$ any $$\epsilon >0$$ . Moreover, known that, 2$$ MaxDBS ...

A unit disk graph is the intersection graph of unit disks in the euclidean plane. We present a polynomial-time approximation scheme for the maximum weight independent set problem in unit disk graphs. In contrast to previously known approximation schemes, our approach does not require a geometric representation (specifying the coordinates of the disk centers). The approximation algorithm present...