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Register a new userAlgorithms and Bounds for Drawing Non-planar Graphs with Crossing-free Subgraphs
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Added by
Giordano Da Lozzo
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing {Gamma} of G in the plane such that the edges of S are not crossed in {Gamma} by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G. Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S; in this setting we discuss different trade-offs between the number of bends and the required drawing area.
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A emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line, a line with slope $-1$. It is an open question to decide in polynomial time whether a given bipartite graph $G$ with bipartition $Acup B$ has a Stick representation where the vertices in $A$ and $B$ correspond to horizontal and vertical segments, respectively. We prove that $G$ has a Stick representation if and only if there are orderings of $A$ and $B$ such that $G$s bipartite adjacency matrix with rows $A$ and columns $B$ excludes three small `forbidden submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of $A$ and $B$ permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of $A$ is given, we present an $O(|A|^3|B|^3)$-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.
A graph drawn in the plane with n vertices is k-fan-crossing free for k > 1 if there are no k+1 edges $g,e_1,...e_k$, such that $e_1,e_2,...e_k$ have a common endpoint and $g$ crosses all $e_i$. We prove a tight bound of 4n-8 on the maximum number of edges of a 2-fan-crossing free graph, and a tight 4n-9 bound for a straight-edge drawing. For k > 2, we prove an upper bound of 3(k-1)(n-2) edges. We also discuss generalizations to monotone graph properties.
We investigate the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes. We insist on straight-line edges and crossing-free drawings. This problem has many connections to other challenging graph-drawing problems such as small-area or small-volume drawings, layered or track drawings, and drawing graphs with low visual complexity. While some facts about our problem are implicit in previous work, this is the first treatment of the problem in its full generality. Our contribution is as follows. We show lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes. In some cases our bounds are asymptotically tight; in some cases we are able to determine exact values. We relate our parameters to standard combinatorial characteristics of graphs (such as the chromatic number, treewidth, maximum degree, or arboricity) and to parameters that have been studied in graph drawing (such as the track number or the number of segments appearing in a drawing). We pay special attention to planar graphs. For example, we show that there are planar graphs that can be drawn in 3-space on a lot fewer lines than in the plane.
An edge guard set of a plane graph $G$ is a subset $Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of edges required to guard any $n$-vertex embedded planar graph $G$: 1- We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that $G$ can be guarded with at most $ frac{2n}{5}$ edges, then extend this approach with a deeper analysis to yield an improved bound of $frac{3n}{8}$ edges for any plane graph. 2- We prove that there exists an edge guard set of $G$ with at most $frac{n}{3}+frac{alpha}{9}$ edges, where $alpha$ is the number of quadrilateral faces in $G$. This improves the previous bound of $frac{n}{3} + alpha$ by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in $G$, we show that $frac{n}{3}$ edges suffice, removing the dependence on $alpha$.
Kuhn, Osthus and Taraz showed that for each gamma>0 there exists C such that any n-vertex graph with minimum degree gamma n contains a planar subgraph with at least 2n-C edges. We find the optimum value of C for all gamma<1/2 and sufficiently large n.
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