Network Discovery and Verification

Beerliová, Zuzana; Eberhard, Felix; Erlebach, Thomas; Hall, Alexander; Hoffmann, Michael M.; Mihaľák, Matúš; Ram, L. Shankar

doi:10.1007/11604686_12

Cited by 12 publications

(1 citation statement)

References 9 publications

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“…The concept of metric dimension of graphs was introduced independently by Slater [16] and Harary and Melter [8]. The applications of metric dimension related to graphs have been studied in combinatorial optimization [15], mastermind game tactics [3], network discovery and verification [1], robot navigation [11] and so on. According to the authors [5], the problem of determining the metric dimension of an arbitrary graph is NP-complete.…”

Section: Metric Dimensionmentioning

confidence: 99%

Some invariants related to threshold and chain graphs

Raja¹,

Wagay²

2024

AMC

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Let G = (V, E) be a finite simple connected graph. We say a graph G realizes a code of the type 0and only if G can be obtained from the code by some rule. Some classes of graphs such as threshold and chain graphs realizes a code of the type mentioned above. The main objective of this research article is to develop some computationally feasible methods to determine some interesting graph theoretical invariants. We present an efficient algorithm to determine the metric dimension of threshold and chain graphs. We compute threshold dimension and restricted threshold dimension of threshold graphs. We discuss L(2, 1)-coloring in threshold and chain graphs. In fact, for every threshold graph G, we establish a formula by which we can obtain the λ-chromatic number of G. Finally, we provide an algorithm to compute the λ-chromatic number of chain graphs.

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Section: Metric Dimensionmentioning

confidence: 99%

Some invariants related to threshold and chain graphs

Raja¹,

Wagay²

2024

AMC

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Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

Damaschke

2010

Networks

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Suppose that we have (i) a graph with an unknown edge set and (ii) access to an oracle that returns all neighbors of a probed vertex. We want to find the hubs, that is, vertices with degree above a given threshold. An almost obvious two-stage randomized strategy is to query randomly sampled vertices and then their neighbors. We prove that this strategy achieves, in certain classes of sparse graphs, an asymptotically optimal query number among all possible querying strategies, including adaptive strategies. A detail of importance for an optimal constant factor is the number of probes in the neighborhood of a hub candidate. We show that 2 is in general the best choice in the graphs of interest. In a similar way we analyze, in the same model, the problem of finding small, almost dominating sets. The problems and graph classes are motivated mainly by the elucidation of interaction networks in molecular biology.

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Network Discovery and Verification with Distance Queries

Erlebach

Hall

Hoffmann

et al. 2006

Lecture Notes in Computer Science

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The network discovery (verification) problem asks for a minimum subset Q ⊆ V of queries in an undirected graph G = (V, E) such that these queries discover all edges and non-edges of the graph. This is motivated by the common approach of combining local measurements in order to obtain maps of the Internet or other dynamically growing networks. In the distance query model, a query at node q returns the distances from q to all other nodes in the graph. We describe how the existence of an individual edge or non-edge in G can be deduced by potentially combining the results of several queries. This leads to a characterization of when a set of queries Q "discovers" the graph G. In the on-line network discovery problem, the graph is initially unknown, and the algorithm has to select queries one by one based only on the results of the previous ones. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O( √ n log n) for graphs on n nodes. We also show lower bounds Ω( √ n) and Ω(log n) on competitive ratios of deterministic on-line algorithms and randomized on-line algorithms, respectively. In the off-line network verification problem, the graph is known in the beginning and the problem asks for a minimum number of queries to verify all edges and non-edges. We show that the problem is N P-hard and present an O(log n)-approximation algorithm.

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Network Discovery and Verification

Cited by 12 publications

References 9 publications

Some invariants related to threshold and chain graphs

Some invariants related to threshold and chain graphs

Finding hidden hubs and dominating sets in sparse graphs by randomized neighborhood queries

Network Discovery and Verification with Distance Queries

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