The good, the bad, and the great: Homomorphisms and cores of random graphs

Bonato, Anthony; Prałat, Paweł

doi:10.1016/j.disc.2008.03.026

Cited by 2 publications

(3 citation statements)

References 7 publications

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“…In order to prove the main result of this paper-Theorem 3.1-we need several lemmas, collected together in the following result. This extends Lemma 1 in [3] to random digraphs.…”

Section: Asymptotic Properties Of Random Digraphssupporting

confidence: 69%

On random digraphs and cores

Parsa,

Kayll

2021

Preprint

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An acyclic homomorphism of a digraph C to a digraph D is a function ρ : V (C) → V (D) such that for every arc uv of C, either ρ(u) = ρ(v), or ρ(u)ρ(v) is an arc of D and for every vertex v ∈ V (D), the subdigraph of C induced by ρ −1 (v) is acyclic. A digraph D is a core if the only acyclic homomorphisms of D to itself are automorphisms. In this paper, we prove that for certain choices of p(n), random digraphs D ∈ D(n, p(n)) are asymptotically almost surely cores. For digraphs, this mirrors a result from [A. Bonato and P. Pra lat, The good, the bad, and the great: homomorphisms and cores of random graphs, Discrete Math., 309 (2009), no. 18, 5535-5539; MR2567955] concerning random graphs and cores.

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“…In order to prove the main result of this paper-Theorem 3.1-we need several lemmas, collected together in the following result. This extends Lemma 1 in [3] to random digraphs.…”

Section: Asymptotic Properties Of Random Digraphssupporting

confidence: 69%

On random digraphs and cores

Parsa,

Kayll

2021

Preprint

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“…The EXACT-WEIGHT-H problem asks, given an edge-weighted graph G, whether there exists a subgraph of G that has weight zero and is isomorphic to H. We say H is a core if every homomorphism from H to H is also an automorphism. Cores are a rich class of graphs, including cliques, odd cycles, and (with high probability) any binomial random graph G(n, p) with edge probability n −1/3 log 2 n < p < 1 − n 1/3 log 2 n (see [10,Theorem 2]). Corollary 6 generalises to EXACT-WEIGHT-H whenever H is a core.…”

Section: Exact-weight K-clique and Other Subgraph Problemsmentioning

confidence: 99%

“…By the correctness of Coarse (Lemma 18) and a union bound over all 1 ≤ i ≤ |L| and all j ∈ [t i ], we have P(E 2 ) ≥ 1 − δ/2. By a union bound with (10), we therefore have…”

Section: 2mentioning

confidence: 99%

Approximately counting and sampling small witnesses using a colourful decision oracle

Dell¹,

Lapinskas²,

Meeks³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the finegrained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691).

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The good, the bad, and the great: Homomorphisms and cores of random graphs

Cited by 2 publications

References 7 publications

On random digraphs and cores

On random digraphs and cores

Approximately counting and sampling small witnesses using a colourful decision oracle

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