6 research outputs found
APUD(1,1) Recognition in Polynomial Time
A unit disk graph is the intersection graph of a set of disk of unit radius
in the Euclidean plane. In 1998, Breu and Kirkpatrick showed that the
recognition problem for unit disk graphs is NP-hard. Given horizontal and
vertical lines, an APUD() is a unit disk graph such that each unit
disk is centered either on a given horizontal or vertical line.
\c{C}a\u{g}{\i}r{\i}c{\i} showed in 2020 that APUD() recognition is
NP-hard. In this paper, we show that APUD() recognition is polynomial time
solvable
Beyond circular-arc graphs -- recognizing lollipop graphs and medusa graphs
In 1992 Bir\'{o}, Hujter and Tuza introduced, for every fixed connected graph
, the class of -graphs, defined as the intersection graphs of connected
subgraphs of some subdivision of . Recently, quite a lot of research has
been devoted to understanding the tractability border for various computational
problems, such as recognition or isomorphism testing, in classes of -graphs
for different graphs . In this work we undertake this research topic,
focusing on the recognition problem. Chaplick, T\"{o}pfer, Voborn\'{\i}k, and
Zeman showed, for every fixed tree , a polynomial-time algorithm recognizing
-graphs. Tucker showed a polynomial time algorithm recognizing -graphs
(circular-arc graphs). On the other hand, Chaplick at al. showed that
recognition of -graphs is -hard if contains two different cycles
sharing an edge.
The main two results of this work narrow the gap between the -hard and
cases of -graphs recognition. First, we show that recognition of
-graphs is -hard when contains two different cycles. On the other
hand, we show a polynomial-time algorithm recognizing -graphs, where is
a graph containing a cycle and an edge attached to it (-graphs are called
lollipop graphs). Our work leaves open the recognition problems of -graphs
for every unicyclic graph different from a cycle and a lollipop. Other
results of this work, which shed some light on the cases that remain open, are
as follows. Firstly, the recognition of -graphs, where is a fixed
unicyclic graph, admits a polynomial time algorithm if we restrict the input to
graphs containing particular holes (hence recognition of -graphs is probably
most difficult for chordal graphs). Secondly, the recognition of medusa graphs,
which are defined as the union of -graphs, where runs over all unicyclic
graphs, is -complete
Bouncing robots in rectilinear polygons
Abstract
In this paper, we describe a bouncing strategy (smart strategy) for a mobile robot that uses one bit of memory for feedback, and guarantees that the robot will traverse all the rooms (and doorways) of a 2D environment. The environment is modeled as a rectilinear polygon (also called orthogonal polygon), and the rooms and the doorways are defined by the decomposition algorithm we describe. Such a decomposition helps the robot to not go back to a room after leaving. We also define the notion of “virtual doors” that have the ability to let the robot through, or make the robot bounce from them. We compared three different types of bouncing rules: smart, random, billiard. The smart strategy grantees to reach to target. Although the random strategy on average behaves the same as the smart strategy, there are rectilinear polygons in which the robot cannot reach the target in the expected time steps. On the other hand, the billiard bouncing strategy can cause the robot to become trapped