18 research outputs found
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
Parameterized Complexity of 1-Planarity
We consider the problem of finding a 1-planar drawing for a general graph,
where a 1-planar drawing is a drawing in which each edge participates in at
most one crossing. Since this problem is known to be NP-hard we investigate the
parameterized complexity of the problem with respect to the vertex cover
number, tree-depth, and cyclomatic number. For these parameters we construct
fixed-parameter tractable algorithms. However, the problem remains NP-complete
for graphs of bounded bandwidth, pathwidth, or treewidth.Comment: WADS 201
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
Orientable biembeddings of cyclic Steiner triple systems from current assignments on Möbius ladder graphs
We give a characterization of a current assignment on the bipartite Möbius ladder graph with 2n+1 rungs. Such an assignment yields an index one current graph with current group that generates an orientable face 2-colorable triangular embedding of the complete graph K12n+7 or, equivalently, an orientable biembedding of two cyclic Steiner triple systems of order 12n+7. We use our characterization to construct Skolem sequences that give rise to such current assignments. These produce many nonisomorphic orientable biembeddings of cyclic Steiner triple systems of order 12n+7
On the minimal nonzero distance between triangular embeddings of a complete graph
Given two triangular embeddings f and f′ of a complete graph K and given a bijection : V(K)→V(K), denote by M() the set of faces (x,y,z) of f such that ((x),(y),(z)) is not a face of f′. The distance between f and f′ is the minimal value of |M()| as ranges over all bijections between the vertices of K. We consider the minimal nonzero distance between two triangular embeddings f and f′ of a complete graph. We show that 4 is the minimal nonzero distance in the case when f and f′ are both nonorientable, and that 6 is the minimal nonzero distance in each of the cases when f and f′ are orientable, and when f is orientable and f′ is nonorientable
Nonorientable biembeddings of Steiner triple systems
Constructions due to Ringel show that there exists a nonorientable face 2-colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all n≡3 (mod 6) with n9. We prove the corresponding existence theorem for n≡1 (mod 6) with n13
Small surface trades in triangular embeddings
We enumerate all possible trades which involve up to six faces of the face set of a triangular embedding of a simple connected graph. These are classified by the underlying combinatorial trade on the associated block design, and by the geometrical arrangement of the faces necessary to avoid creation of a pseudosurface in the trading operation. The relationship of each of these trades to surface orientability is also established
Locally 2-dimensional Sperner problems complete for the polynomial parity argument classes
In this paper, we define three Sperner problems on specific surfaces and prove that they are complete respectively for the classes PPAD, PPADS and PPA. This is the first time that locally 2-dimensional Sperner problems are proved to be complete for any of the polynomial parity argument classes.
Tiziana Calamoneri, Irene Finocchi, Giuseppe F. Italiano (Eds.): CIAC 2006. 6th Italian conference on algorithms and complexity. Rome, 2006. Berlin, Springer, 2006