Incremental Strong Connectivity and 2-Connectivity in Directed Graphs

Georgiadis, Loukas; Italiano, Giuseppe F.; Parotsidis, Nikos

doi:10.1007/978-3-319-77404-6_39

Cited by 6 publications

(12 citation statements)

References 63 publications

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“…Third, the dynamic maintenance of 2-connectivity properties in directed graphs deserves further attention. After this work, we have been able to apply the algorithmic framework developed in this paper to the incremental maintenance of the 2-edge-and the 2-vertex-connected components of directed graphs [21,22]. The decremental version of these problems, where we wish to maintain the 2-edge-and the 2-vertex-connected components of a directed graph under edge deletions, were considered in [16] using different techniques.…”

Section: Discussionmentioning

confidence: 99%

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Strong Connectivity in Directed Graphs under Failures, with Applications

Georgiadis¹,

Italiano²,

Parotsidis³

2020

SIAM J. Comput.

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In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let G be a digraph with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. As a first result, we show how to compute in O(m + n) worst-case time: • The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. • The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). As a second set of results, we show how to build in O(m + n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: • Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v). • Test whether an edge or a vertex separates two query vertices in O(1) worst-case time. • Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the 1-connectivity (i.e., 1-edge and 1-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts.

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Section: Discussionmentioning

confidence: 99%

“…We believe that the framework developed in this paper may be of independent interest and may prove to be useful for other problems as well. In particular, after this work, we have been able to apply it to the incremental maintenance of 2-edge connectivity properties on digraphs [21,22].…”

Section: Introductionmentioning

confidence: 99%

Strong Connectivity in Directed Graphs under Failures, with Applications

Georgiadis¹,

Italiano²,

Parotsidis³

2020

SIAM J. Comput.

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“…SCC decomposition alone leads to Kirchhoff polynomial factorisation which is not guaranteed to be prime. Yet we include it as a sub-heuristic due to recent results in strong connectivity allowing to retrieve all strong bridges (4), the total number of SCCs, and the size of the largest and of the smallest SCCs obtained after edge deletion in linear time (5). Note that, in order to have comparable running times when decomposing into prime components and SCCs, we naively delete-contract each considered edge and do not employ the mentioned recent advancements.…”

Section: Description Of Used Heuristicsmentioning

confidence: 99%

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks

Yordanov

Stelling

2020

J. R. Soc. Interface.

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Kirchhoff polynomials are central for deriving symbolic steady-state expressions of models whose dynamics are governed by linear diffusion on graphs. In biology, such models have been unified under a common linear framework subsuming studies across areas such as enzyme kinetics, G-protein coupled receptors, ion channels and gene regulation. Due to ‘history dependence’ away from thermodynamic equilibrium, these models suffer from a (super) exponential growth in the size of their symbolic steady-state expressions and, respectively, Kirchhoff polynomials. This algebraic explosion has limited applications of the linear framework. However, recent results on the graph-based prime factorization of Kirchhoff polynomials may help subdue the combinatorial complexity. By prime decomposing the graphs contained in an expression of Kirchhoff polynomials and identifying the graphs giving rise to equal polynomials, we formulate a coarse-grained variant of the expression suitable for symbolic simplification. We devise criteria to efficiently test the equality of Kirchhoff polynomials and propose two heuristic algorithms to explicitly generate individual Kirchhoff polynomials in a compressed form; they are inspired by algebraic simplifications but operate on the corresponding graphs. We illustrate the practicality of the developed theory and algorithms for a diverse set of graphs of different sizes and for non-equilibrium gene regulation analyses.

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“…We hereinafter refer to such heuristic as the Critical Node Hueuristic (CNH). The CNH is based on the framework proposed in [19] which uses the notions of Dominators [9], Strong Articulation Points [26] and Loop Nesting Forest [47] to build a novel data structure which stores the size of all Strongly Connected Component created by each node removal.…”

Section: B Critical Nodesmentioning

confidence: 99%

Critical nodes reveal peculiar features of human essential genes and protein interactome

Celestini

Cianfriglia

Mastrostefano

et al. 2019

Preprint

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Network-based ranking methods (e.g., centrality analysis) have found extensive use in systems biology and network medicine for the prediction of essential proteins, for the prioritization of drug targets candidates in the treatment of several pathologies and in biomarker discovery, and for human disease genes identification. We here studied the connectivity of the human protein-protein interaction network (i.e., the interactome) to find the nodes whose removal has the heaviest impact on the network, i.e., maximizes its fragmentation. Such nodes are known as Critical Nodes (CNs). Specifically, we implemented a Critical Node Heuristic (CNH) and compared its performance against other four heuristics based on well known centrality measures. To better understand the structure of the interactome, the CNs' role played in the network, and the different heuristics' capabilities to grasp biologically relevant nodes, we compared the sets of nodes identified as CNs by each heuristic with two experimentally validated sets of essential genes, i.e., the genes whose removal impact on a given organism's ability to survive. Our results show that classical centrality measures (i.e., closeness centrality, degree) found more essential genes with respect to CNH on the current version of the human interactome, however the removal of such nodes does not have the greatest impact on interactome connectivity, while, interestingly, the genes identified by CNH show peculiar characteristics both from the topological and the biological point of view. Finally, even if a relevant fraction of essential genes is found via the classical centrality measures, the same measures seem to fail in identifying the whole set of essential genes, suggesting once again that some of them are not central in the network, that there may be biases in the current interaction data, and that different, combined graph theoretical and other techniques should be applied for their discovery.

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Incremental Strong Connectivity and 2-Connectivity in Directed Graphs

Cited by 6 publications

References 63 publications

Strong Connectivity in Directed Graphs under Failures, with Applications

Strong Connectivity in Directed Graphs under Failures, with Applications

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks

Critical nodes reveal peculiar features of human essential genes and protein interactome

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