38,152 research outputs found
Beyond Outerplanarity
We study straight-line drawings of graphs where the vertices are placed in
convex position in the plane, i.e., convex drawings. We consider two families
of graph classes with nice convex drawings: outer -planar graphs, where each
edge is crossed by at most other edges; and, outer -quasi-planar graphs
where no edges can mutually cross. We show that the outer -planar graphs
are -degenerate, and consequently that every
outer -planar graph can be -colored, and this
bound is tight. We further show that every outer -planar graph has a
balanced separator of size . This implies that every outer -planar
graph has treewidth . For fixed , these small balanced separators
allow us to obtain a simple quasi-polynomial time algorithm to test whether a
given graph is outer -planar, i.e., none of these recognition problems are
NP-complete unless ETH fails. For the outer -quasi-planar graphs we prove
that, unlike other beyond-planar graph classes, every edge-maximal -vertex
outer -quasi planar graph has the same number of edges, namely . We also construct planar 3-trees that are not outer
-quasi-planar. Finally, we restrict outer -planar and outer
-quasi-planar drawings to \emph{closed} drawings, where the vertex sequence
on the boundary is a cycle in the graph. For each , we express closed outer
-planarity and \emph{closed outer -quasi-planarity} in extended monadic
second-order logic. Thus, closed outer -planarity is linear-time testable by
Courcelle's Theorem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
The edge chromatic number of outer-1-planar graphs
A graph is outer-1-planar if it can be drawn in the plane so that all
vertices are on the outer face and each edge is crossed at most once. In this
paper, we completely determine the edge chromatic number of outer 1-planar
graphs
On the Recognition of Fan-Planar and Maximal Outer-Fan-Planar Graphs
Fan-planar graphs were recently introduced as a generalization of 1-planar
graphs. A graph is fan-planar if it can be embedded in the plane, such that
each edge that is crossed more than once, is crossed by a bundle of two or more
edges incident to a common vertex. A graph is outer-fan-planar if it has a
fan-planar embedding in which every vertex is on the outer face. If, in
addition, the insertion of an edge destroys its outer-fan-planarity, then it is
maximal outer-fan-planar. In this paper, we present a polynomial-time algorithm
to test whether a given graph is maximal outer-fan-planar. The algorithm can
also be employed to produce an outer-fan-planar embedding, if one exists. On
the negative side, we show that testing fan-planarity of a graph is NP-hard,
for the case where the rotation system (i.e., the cyclic order of the edges
around each vertex) is given
Bar 1-Visibility Drawings of 1-Planar Graphs
A bar 1-visibility drawing of a graph is a drawing of where each
vertex is drawn as a horizontal line segment called a bar, each edge is drawn
as a vertical line segment where the vertical line segment representing an edge
must connect the horizontal line segments representing the end vertices and a
vertical line segment corresponding to an edge intersects at most one bar which
is not an end point of the edge. A graph is bar 1-visible if has a bar
1-visibility drawing. A graph is 1-planar if has a drawing in a
2-dimensional plane such that an edge crosses at most one other edge. In this
paper we give linear-time algorithms to find bar 1-visibility drawings of
diagonal grid graphs and maximal outer 1-planar graphs. We also show that
recursive quadrangle 1-planar graphs and pseudo double wheel 1-planar graphs
are bar 1-visible graphs.Comment: 15 pages, 9 figure
The structure and the list 3-dynamic coloring of outer-1-planar graphs
An outer-1-planar graph is a graph admitting a drawing in the plane so that
all vertices appear in the outer region of the drawing and every edge crosses
at most one other edge. This paper establishes the local structure of
outer-1-planar graphs by proving that each outer-1-planar graph contains one of
the seventeen fixed configurations, and the list of those configurations is
minimal in the sense that for each fixed configuration there exist
outer-1-planar graphs containing this configuration that do not contain any of
another sixteen configurations. There are two interesting applications of this
structural theorem. First of all, we conclude that every (resp.maximal)
outer-1-planar graph of minimum degree at least 2 has an edge with the sum of
the degrees of its two end-vertices being at most 9 (resp.7), and this upper
bound is sharp. On the other hand, we show that the list 3-dynamic chromatic
number of every outer-1-planar graph is at most 6, and this upper bound is best
possible
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