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83
Succinct Representation of Balanced Parentheses, Static Trees and Planar Graphs
, 1999
"... We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set ..."
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Cited by 140 (9 self)
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We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set of navigational operations than has previously been considered in similar work. In the case of binary trees, for instance, we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned. The approach is applied to produce succinct representation of planar graphs in which one can test adjacency in constant time.
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 32 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 30 (18 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Compact Routing Tables for Graphs of Bounded Genus
, 2000
"... This paper deals with compact shortest path routing tables on weighted graphs with n nodes. For planar graphs we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log 2+ n) bitoperations per node to extract the route, for any constant > ..."
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Cited by 30 (12 self)
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This paper deals with compact shortest path routing tables on weighted graphs with n nodes. For planar graphs we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log 2+ n) bitoperations per node to extract the route, for any constant > 0. We obtain the same bounds for graphs of crossingedge number bounded by o(n= log n), and we generalize for graphs of genus bounded by > 0 yielding a size of n log +O(n) bits per node. Actually we prove a sharp upper bound of 2n log k +O(n) for graphs of pagenumber k, and a lower bound of n log k o(n log k) bits. These results are obtained by the use of dominating sets, compact coding of noncrossing partitions, and kpage representation of graphs.
On Simultaneous Planar Graph Embeddings
 COMPUT. GEOM
, 2003
"... We consider the problem of simultaneous embedding of planar graphs. There are two variants ..."
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Cited by 29 (8 self)
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We consider the problem of simultaneous embedding of planar graphs. There are two variants
Geometric Thickness of Complete Graphs
 J. GRAPH ALGORITHMS APPL
, 2000
"... We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straightline edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantiti ..."
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Cited by 28 (4 self)
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We define the geometric thickness of a graph to be the smallest number of layers such that we can draw the graph in the plane with straightline edges and assign each edge to a layer so that no two edges on the same layer cross. The geometric thickness lies between two previously studied quantities, the (graphtheoretical) thickness and the book thickness. We investigate the geometric thickness of the family of complete graphs, {Kn}. We show that the geometric thickness of Kn lies between #(n/5.646) + 0.342# and #n/4#, and we give exact values of the geometric thickness of Kn for n # 12 and n #{15, 16}. We also consider the geometric thickness of the family of complete bipartite graphs. In particular, we show that, unlike the case of complete graphs, there are complete bipartite graphs with arbitrarily large numbers of vertices for which the geometric thickness coincides with the standard graphtheoretical thickness.
Stack And Queue Layouts Of Directed Acyclic Graphs: Part I
, 1996
"... . Stack layouts and queue layouts of undirected graphs have been used to model problems in fault tolerant computing and in parallel process scheduling. However, problems in parallel process scheduling are more accurately modeled by stack and queue layouts of directed acyclic graphs (dags). A stack ..."
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Cited by 26 (3 self)
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. Stack layouts and queue layouts of undirected graphs have been used to model problems in fault tolerant computing and in parallel process scheduling. However, problems in parallel process scheduling are more accurately modeled by stack and queue layouts of directed acyclic graphs (dags). A stack layout of a dag is similar to a stack layout of an undirected graph, with the additional requirement that the nodes of the dag be in some topological order. A queue layout is defined in an analogous manner. The stacknumber (queuenumber) of a dag is the smallest number of stacks (queues) required for its stack layout (queue layout). In this paper, bounds are established on the stacknumber and queuenumber of two classes of dags: tree dags and unicyclic dags. In particular, any tree dag can be laid out in 1 stack and in at most 2 queues; and any unicyclic dag can be laid out in at most 2 stacks and in at most 2 queues. Forbidden subgraph characterizations of 1queue tree dags and 1queue cycle d...
Layout of Graphs with Bounded TreeWidth
 2002, submitted. Stacks, Queues and Tracks: Layouts of Graph Subdivisions 41
, 2004
"... A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a gr ..."
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Cited by 25 (19 self)
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A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queuenumber. A threedimensional (straight line grid) drawing of a graph represents the vertices by points in Z and the edges by noncrossing linesegments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of threedimensional drawing of a graph G is closely related to the queuenumber of G. In particular, if G is an nvertex member of a proper minorclosed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queuenumber.
Stack And Queue Layouts Of Posets
 SIAM J. Discrete Math
, 1995
"... . The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower ..."
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Cited by 19 (4 self)
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. The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower bound of \Omega\Gamma p n) is shown for the queuenumber of the class of nelement planar posets. The queuenumber of a planar poset is shown to be within a small constant factor of its width. The stacknumber of nelement posets with planar covering graphs is shown to be \Theta(n). These results exhibit sharp differences between the stacknumber and queuenumber of posets as well as between the stacknumber (queuenumber) of a poset and the stacknumber (queuenumber) of its covering graph. Key words. poset, queue layout, stack layout, book embedding, Hasse diagram, jumpnumber AMS subject classifications. 05C99, 68R10, 94C15 1. Introduction. Stack and queue layouts of undirected graphs appear ...