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Graph center

About: Graph center is a(n) research topic. Over the lifetime, 948 publication(s) have been published within this topic receiving 19710 citation(s).

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Papers
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Journal ArticleDOI: 10.1137/0136016
Abstract: Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.

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Topics: Vertex separator (67%), Neighbourhood (graph theory) (65%), Wheel graph (61%) ...read more

1,283 Citations


Open accessJournal ArticleDOI: 10.1002/RSA.1009
Abstract: Recently, Barabasi and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd−γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n1/15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001

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Topics: Path graph (63%), Wheel graph (62%), Graph center (62%) ...read more

848 Citations


Open accessJournal ArticleDOI: 10.1016/0095-8956(80)90074-X
George J. Minty1Institutions (1)
Abstract: A graph is claw-free if: whenever three (distinct) vertices are joined to a single vertex, those three vertices are a nonindependent (nonstable) set. Given a finite claw-free graph with real numbers (weights) assigned to the vertices, we exhibit an algorithm for producing an independent set of vertices of maximum total weight. This algorithm is “efficient” in the sense of J. Edmonds, that is to say, the number of computational steps required is of polynomial (not exponential or factorial) order in n , the number of vertices of the graph. This problem was solved earlier by Edmonds for the special case of “edge-graphs”; our solution is by reducing the more general problem to the earlier-solved special case. Separate attention is given to the case in which all weights are (+1) and thus an independent set is sought which is maximal in the sense of its cardinality.

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Topics: Graph center (71%), Independent set (70%), Neighbourhood (graph theory) (68%) ...read more

468 Citations


Open accessJournal ArticleDOI: 10.1109/18.661507
Abstract: We investigate a new class of codes for the optimal covering of vertices in an undirected graph G such that any vertex in G can be uniquely identified by examining the vertices that cover it. We define a ball of radius t centered on a vertex /spl upsi/ to be the set of vertices in G that are at distance at most t from /spl upsi/. The vertex /spl upsi/ is then said to cover itself and every other vertex in the ball with center /spl upsi/. Our formal problem statement is as follows: given an undirected graph G and an integer t/spl ges/1, find a (minimal) set C of vertices such that every vertex in G belongs to a unique set of balls of radius t centered at the vertices in C. The set of vertices thus obtained constitutes a code for vertex identification. We first develop topology-independent bounds on the size of C. We then develop methods for constructing C for several specific topologies such as binary cubes, nonbinary cubes, and trees. We also describe the identification of sets of vertices using covering codes that uniquely identify single vertices. We develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords. Finally, we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems.

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  • TABLE IV NUMBER OF CODEWORDS IN AN OPTIMAL CODE FOR IDENTIFYING SETS OF VERTICES WITH CARDINALITY UP TO TWO
    TABLE IV NUMBER OF CODEWORDS IN AN OPTIMAL CODE FOR IDENTIFYING SETS OF VERTICES WITH CARDINALITY UP TO TWO
  • Fig. 5. An optimal graph for uniquely identifying a single vertex.
    Fig. 5. An optimal graph for uniquely identifying a single vertex.
  • TABLE III NUMBER OF CODEWORDSm(1) FOR (a) BALANCED BINARY TREE (P = 2); (b) BALANCED TERNARY TREE (p = 3)
    TABLE III NUMBER OF CODEWORDSm(1) FOR (a) BALANCED BINARY TREE (P = 2); (b) BALANCED TERNARY TREE (p = 3)
  • Fig. 3. Codewords (shaded) with= 1 for a (a) hexagonal mesh and (b) triangular mesh (the ends wrap around).
    Fig. 3. Codewords (shaded) with= 1 for a (a) hexagonal mesh and (b) triangular mesh (the ends wrap around).
Topics: Neighbourhood (graph theory) (70%), Graph center (66%), Feedback vertex set (64%) ...read more

437 Citations


Open accessJournal ArticleDOI: 10.1016/J.TCS.2005.10.007
Dániel Marx1Institutions (1)
Abstract: We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given l terminals is separated from the others, (b) each of the given l pairs of terminals is separated, (c) exactly l vertices are cut away from the graph, (d) exactly l connected vertices are cut away from the graph, (e) the graph is separated into at least l components. We show that if both k and l are parameters, then (a), (b) and (d) are fixed-parameter tractable, while (c) and (e) are W[1]-hard.

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Topics: Path graph (67%), Graph power (65%), Graph center (65%) ...read more

290 Citations


Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20211
20201
20192
20186
201738
201659

Top Attributes

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Topic's top 5 most impactful authors

Satoshi Taoka

5 papers, 29 citations

Michael A. Henning

5 papers, 130 citations

Ralph J. Faudree

4 papers, 31 citations

Serge Gaspers

4 papers, 31 citations

David Peleg

4 papers, 291 citations

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