Results 1  10
of
35
Minimal EdgeCoverings of Pairs of Sets
, 1995
"... A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and ..."
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Cited by 59 (13 self)
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A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph kedgeconnected. As another consequence, we solve the corresponding nodeconnectivity augmentation problem in directed graphs.
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 31 (19 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...
Minimum Bounded Degree Spanning Trees
, 2006
"... We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. Thi ..."
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Cited by 30 (0 self)
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We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to ledgeconnected subgraphs is briefly discussed.
The Cycle Space of an Infinite Graph
 COMB., PROBAB. COMPUT
, 2004
"... Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph togethe ..."
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Cited by 26 (9 self)
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Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach
Applications Of Submodular Functions
, 1993
"... Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization. ..."
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Cited by 24 (2 self)
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Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.
Sublogarithmic Distributed MIS Algorithm for Sparse Graphs using NashWilliams Decomposition
 In Journal of Distributed Computing Special Issue of selected papers from PODC
, 2008
"... We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on gr ..."
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Cited by 15 (2 self)
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We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graphtheoretic structure, called NashWilliams forestsdecomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearlytight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.
Acyclic, star and oriented colourings of graph subdivisions
 Discrete Math. Theoret. Comput. Sci
, 2005
"... Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This pa ..."
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Cited by 13 (5 self)
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Let G be a graph with chromatic number χ(G). A vertex colouring of G is acyclic if each bichromatic subgraph is a forest. A star colouring of G is an acyclic colouring in which each bichromatic subgraph is a star forest. Let χa(G) and χs(G) denote the acyclic and star chromatic numbers of G. This paper investigates acyclic and star colourings of subdivisions. Let G ′ be the graph obtained from G by subdividing each edge once. We prove that acyclic (respectively, star) colourings of G ′ correspond to vertex partitions of G in which each subgraph has small arboricity (chromatic index). It follows that χa(G ′), χs(G ′ ) and χ(G) are tied, in the sense that each is bounded by a function of the other. Moreover the binding functions that we establish are all tight. The oriented chromatic number − → χ (G) of an (undirected) graph G is the maximum, taken over all orientations D of G, of the minimum number of colours in a vertex colouring of D such that between any two colour classes, all edges have the same direction. We prove that − → χ (G ′ ) = χ(G) whenever χ(G) ≥ 9.
Characterizations of arboricity of graphs
 Ars Combinatorica
"... The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spannin ..."
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Cited by 12 (0 self)
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The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of trees or forests dates back to NashWilliams and Tutte [6, 7, 11]. Since then, many authors have examined various tree decompositions of classes of graphs (for example [2, 8]). The aim of this paper is to give several characterizations for
Rectanglevisibility Layouts of Unions and Products of Trees
 J. Graph Algorithms Appl
, 1998
"... The paper considers representations of unions and products of trees as rectanglevisibility graphs (abbreviated RVGs), i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. Our main results are that the union of any tree (or forest ..."
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Cited by 6 (3 self)
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The paper considers representations of unions and products of trees as rectanglevisibility graphs (abbreviated RVGs), i.e., graphs whose vertices are rectangles in the plane, with adjacency determined by horizontal and vertical visibility. Our main results are that the union of any tree (or forest) with a depth1 tree is an RVG, and that the union of two depth2 trees and the union of a depth3 tree with a matching are subgraphs of RVGs. We also show that the cartesian product of two forests is an RVG. Communicated by J. S. B. Mitchell: submitted January 1998; revised September 1998. A. Dean and J. Hutchinson, RVG Layouts, JGAA, 2(8) 121 (1998) 2 1 Introduction In this paper we study aspects of the question of how to represent a graph in the plane as a rectanglevisibility graph (RVG for short). In such a representation the vertices are drawn as rectangles with horizontal and vertical sides, and two vertices are adjacent if and only if their rectangles can be connected by a horiz...
Locally finite graphs with ends: a topological approach
"... This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. Thi ..."
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Cited by 6 (6 self)
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This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open