Non-crossing Shortest Paths in Undirected Unweighted Planar Graphs in Linear Time

Balzotti, Lorenzo; Franciosa, Paolo Giulio

doi:10.1007/978-3-031-09574-0_6

Cited by 5 publications

(4 citation statements)

References 29 publications

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“…In the case of integer capacity values that do not exceed a small constant, or in the more general case in which capacity values are integers with bounded sum we also prove the following corollaries by using the results in [8, 9, 34, 51].…”

Section: Introductionmentioning

confidence: 99%

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How vulnerable is an undirected planar graph with respect to max flow

Balzotti,

Franciosa

2024

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We study the problem of computing the vitality of edges and vertices with respect to the ‐max flow in undirected planar graphs, where the vitality of an edge/vertex is the ‐max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the ‐max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute ‐max flow . We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in space.

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Section: Introductionmentioning

confidence: 99%

“…FIGURE9 (A) The graph U in bold and (B) subgraphs Left i and Right i are highlighted. (C) Subgraph Ω i,j , for some i < j.…”

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confidence: 99%

How vulnerable is an undirected planar graph with respect to max flow

Balzotti,

Franciosa

2024

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“…, p k , where, for each i 2 [k], p i is a shortest path from x i to y i in G, x i and y i are on the infinite face f 1 of G, p i and p j are non-crossing for every i, j 2 [k]. We stress that we know U but we ignore the p i 's; indeed, algorithms solving the NCSP problem in [3,4,21] compute the union of the non-crossing shortest paths without listing every single path.…”

Section: Introductionmentioning

confidence: 99%

“…10. (a) the union of shortest i-paths, for i 2 [4], in unit-weighted graph, each different path has different style, (b) the union of{ i } i2[4] , the output paths of algorithm ImplicitPaths.…”

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confidence: 99%