7,925 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
Recognizing Weighted Disk Contact Graphs
Disk contact representations realize graphs by mapping vertices bijectively
to interior-disjoint disks in the plane such that two disks touch each other if
and only if the corresponding vertices are adjacent in the graph. Deciding
whether a vertex-weighted planar graph can be realized such that the disks'
radii coincide with the vertex weights is known to be NP-hard. In this work, we
reduce the gap between hardness and tractability by analyzing the problem for
special graph classes. We show that it remains NP-hard for outerplanar graphs
with unit weights and for stars with arbitrary weights, strengthening the
previous hardness results. On the positive side, we present constructive
linear-time recognition algorithms for caterpillars with unit weights and for
embedded stars with arbitrary weights.Comment: 24 pages, 21 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
RECOGNIZING WEIGHTED AND SEEDED DISK GRAPHS
Disk intersection representations realize graphs by mapping vertices bijectively to disks in the plane such that two disks intersect each other if and only if the corresponding vertices are adjacent in the graph. If intersections are restricted to touching points of the boundaries, we call them disk contact representations. Deciding whether a vertex-weighted planar graph can be realized such that the disks\u27 radii coincide with the vertex weights is known to be NP-hard for both contact and intersection representations. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that in the contact scenario it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present a constructive linear-time recognition algorithm for embedded stars with arbitrary weights.
We also consider a version of the problem in which the disks of a representation are supposed to cover preassigned points, called seeds. We show that both for contact and intersection representations this problem is NP-hard for unit weights even if the given graph is a path. If the disks\u27 radii are not prescribed, the problem remains NP-hard for trees in the contact scenario
A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs
We consider the problem of partitioning the set of vertices of a given unit
disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and
various constant factor approximations are known, with the current best ratio
of 3. Our main result is a {\em weakly robust} polynomial time approximation
scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a
clique partition or (ii) gives a certificate that the graph is not a UDG; for
the case (i) that it computes a clique partition, we show that it is guaranteed
to be within (1+\eps) ratio of the optimum if the input is UDG; however if
the input is not a UDG it either computes a clique partition as in case (i)
with no guarantee on the quality of the clique partition or detects that it is
not a UDG. Noting that recognition of UDG's is NP-hard even if we are given
edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be
transformed into an O(\frac{\log^* n}{\eps^{O(1)}}) time distributed PTAS.
We consider a weighted version of the clique partition problem on vertex
weighted UDGs that generalizes the problem. We note some key distinctions with
the unweighted version, where ideas useful in obtaining a PTAS breakdown. Yet,
surprisingly, it admits a (2+\eps)-approximation algorithm for the weighted
case where the graph is expressed, say, as an adjacency matrix. This improves
on the best known 8-approximation for the {\em unweighted} case for UDGs
expressed in standard form.Comment: 21 pages, 9 figure
Max-Cut and Max-Bisection are NP-hard on unit disk graphs
We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk
graphs. We also show that -precision graphs are planar for >
1 / \sqrt{2}$
A PTAS for the minimum dominating set problem in unit disk graphs
We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. \u
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