Torus-like graphs and their paired many-to-many disjoint path covers

Park, Jung-Heum

doi:10.1016/j.dam.2020.09.008

Cited by 12 publications

(9 citation statements)

References 26 publications

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“…The many-to-many k-disjoint path cover problem studied in [10], [15], [16], [17], [23], looks at breaking a graph into k disjoint paths, with given starting and ending vertices (sources and sinks) such that every vertex is in exactly one path. We say it is paired if each source is required to be in the same path as a specified sink, otherwise it is unpaired.…”

Section: Many-to-many K-disjoint Path Cover and Conclusionmentioning

confidence: 99%

“…This means that solution paths between connected pair of cells may need to be non-simple, which adds some additional complexity to the problem of finding a zero-knowledge proof. Flow Free also relates deeply to research on a generalization of the problem of finding a Hamiltonian path with given starting points called the disjoint covering paths problem (studied by [17], [10], [23], [16], and [13] among many others), because the solution to each game can easily be mapped to a disjoint covering path for the underlying adjacency graph of the game. Numberlink corresponds to the question of vertex-disjoint paths that do not necessarily cover every vertex.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Physical Zero-knowledge Proofs for Flow Free, Hamiltonian Cycles, and Many-to-many k-disjoint Covering Paths

Hart¹,

McGinnis²

2022

Preprint

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Section: Many-to-many K-disjoint Path Cover and Conclusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Physical Zero-knowledge Proofs for Flow Free, Hamiltonian Cycles, and Many-to-many k-disjoint Covering Paths

Hart¹,

McGinnis²

2022

Preprint

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“…Definition 2 (see [12]). An m-dimensional torus-like graph, m ≥ 1, is a graph obtained through the cycle-based recursive construction from (m − 1)-dimensional torus-like graphs G 0 , .…”

Section: Introductionmentioning

confidence: 99%

“…Disjoint path cover problems have been studied for various classes of graphs, including recent studies on dense graphs [13], cube of connected graphs [14], balanced hypercubes [15], [16], hypercube-like networks [17], [18], recursive circulants [19], directed graphs [20], k-ary n-cubes [21], and torus networks [22]. In particular, the paired disjoint path cover problem for torus-like graphs was investigated in [12] for a nonbipartite case and in [23] for a bipartite case. In addition, a study on unpaired disjoint path covers of a bipartite k-ary n-cube, which is a special form of torus, can be found in [24].…”

Section: Introductionmentioning

confidence: 99%

Unpaired Many-to-Many Disjoint Path Covers in Nonbipartite Torus-Like Graphs With Faulty Elements

Park

2022

IEEE Access

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One of the key problems in parallel processing is finding disjoint paths in the underlying graph of an interconnection network. The disjoint path cover of a graph is a set of pairwise vertex-disjoint paths that altogether cover every vertex of the graph. Given disjoint source and sink sets, S = {s 1 , . . . , s k } and T = {t 1 , . . . , t k }, in graph G, an unpaired many-to-many k-disjoint path cover joining S and T is a disjoint path cover {P 1 , . . . , P k }, in which each path P i runs from source s i to some sink t j . In this paper, we reveal that a nonbipartite torus-like graph, if built from lower dimensional torus-like graphs that have good disjoint-path-cover properties of the unpaired type, retains such a good property. As a result, an m-dimensional nonbipartite torus, m ≥ 2, with at most f vertex and/or edge faults has an unpaired manyto-many k-disjoint path cover joining arbitrary disjoint sets S and T of size k each, subject to k ≥ 2 and f + k ≤ 2m − 2. The bound of 2m − 2 on f + k is nearly optimal.INDEX TERMS Disjoint path, path cover, path partition, torus, toroidal grid, interconnection network.

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“…The edge-disjoint paths problem is a fundamental problem in networks, consisting of connecting as many demand pairs as possible in a graph via edge-disjoint paths. In disjoint paths problems, instead of considering the paired and unpaired many-to-many disjoint paths cover problem [9], [13], [20], [30], we focus on the problem of evaluation maximum number of many-to-many edge disjoint paths of a graph. Let g be a positive integer.…”

Section: Introductionmentioning

confidence: 99%

Many-to-Many Disjoint Paths in Augmented Cubes With Exponential Fault Edges

Zhang

2021

IEEE Access

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It is common knowledge that edge disjoint paths have close relationship with the edge connectivity. Motivated by the well-known Menger theorem, we find that the maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs with g vertices in G can also define by the minimum modified edge-cut, called the g-extra edge-connectivity of G (λ g (G)). It is the cardinality of the minimum set of edges in G, if such a set exists, whose deletion disconnects G and leaves every remaining component with at least g vertices. The n-dimensional augmented cube AQ n is a variant of hypercube Q n . In this paper, we observe that the g-extra edge-connectivity of the augmented cube for some exponentially large enough g exists a concentration behavior, for about 72.22 percent values of g ≤ 2 n−1 , and that the g-extra edge-connectivity of AQ n (n ≥ 3) concentrates on n 2 − 1 special values. Specifically, we prove that the exact value of g-extra edge-connectivity of augmented cube is a constant 2( n 2 − r)2+r , where n ≥ 3, r = 1, 2, . . . , n 2 − 1 and l r = 2 2r+1 −2 3 if n is odd and l r = 2 2r+2 −4 3 if n is even. The above upper and lower bounds of g are sharp. Moreover, we also obtain the exponential edge disjoint paths in AQ n with edge faults.INDEX TERMS Fault tolerance, many-to-many edge disjoint paths, interconnected networks, exponential fault edges.

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Torus-like graphs and their paired many-to-many disjoint path covers

Cited by 12 publications

References 26 publications

Physical Zero-knowledge Proofs for Flow Free, Hamiltonian Cycles, and Many-to-many k-disjoint Covering Paths

Physical Zero-knowledge Proofs for Flow Free, Hamiltonian Cycles, and Many-to-many k-disjoint Covering Paths

Unpaired Many-to-Many Disjoint Path Covers in Nonbipartite Torus-Like Graphs With Faulty Elements

Many-to-Many Disjoint Paths in Augmented Cubes With Exponential Fault Edges

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