11,358 research outputs found
Contact Representations of Graphs in 3D
We study contact representations of graphs in which vertices are represented
by axis-aligned polyhedra in 3D and edges are realized by non-zero area common
boundaries between corresponding polyhedra. We show that for every 3-connected
planar graph, there exists a simultaneous representation of the graph and its
dual with 3D boxes. We give a linear-time algorithm for constructing such a
representation. This result extends the existing primal-dual contact
representations of planar graphs in 2D using circles and triangles. While
contact graphs in 2D directly correspond to planar graphs, we next study
representations of non-planar graphs in 3D. In particular we consider
representations of optimal 1-planar graphs. A graph is 1-planar if there exists
a drawing in the plane where each edge is crossed at most once, and an optimal
n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a
linear-time algorithm for representing optimal 1-planar graphs without
separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph
admits a representation with boxes. Hence, we consider contact representations
with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a
quadratic-time algorithm for representing optimal 1-planar graph with L-shaped
polyhedra
A Victorian Age Proof of the Four Color Theorem
In this paper we have investigated some old issues concerning four color map
problem. We have given a general method for constructing counter-examples to
Kempe's proof of the four color theorem and then show that all counterexamples
can be rule out by re-constructing special 2-colored two paths decomposition in
the form of a double-spiral chain of the maximal planar graph. In the second
part of the paper we have given an algorithmic proof of the four color theorem
which is based only on the coloring faces (regions) of a cubic planar maps. Our
algorithmic proof has been given in three steps. The first two steps are the
maximal mono-chromatic and then maximal dichromatic coloring of the faces in
such a way that the resulting uncolored (white) regions of the incomplete
two-colored map induce no odd-cycles so that in the (final) third step four
coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio
Nested hierarchies in planar graphs
We construct a partial order relation which acts on the set of 3-cliques of a
maximal planar graph G and defines a unique hierarchy. We demonstrate that G is
the union of a set of special subgraphs, named `bubbles', that are themselves
maximal planar graphs. The graph G is retrieved by connecting these bubbles in
a tree structure where neighboring bubbles are joined together by a 3-clique.
Bubbles naturally provide the subdivision of G into communities and the tree
structure defines the hierarchical relations between these communities
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
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