---
_id: '11894'
abstract:
- lang: eng
  text: "Graph sparsification aims at compressing large graphs into smaller ones while
    preserving important characteristics of the input graph. In this work we study
    vertex sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices.
    We focus on the following notions: (1) Given a digraph \U0001D43A=(\U0001D449,\U0001D438)
    and terminal vertices \U0001D43E⊂\U0001D449 with |\U0001D43E|=\U0001D458, a (vertex)
    reachability sparsifier of \U0001D43A is a digraph \U0001D43B=(\U0001D449\U0001D43B,\U0001D438\U0001D43B),
    \U0001D43E⊂\U0001D449\U0001D43B that preserves all reachability information among
    terminal pairs. Let |\U0001D449\U0001D43B| denote the size of \U0001D43B. In this
    work we introduce the notion of reachability-preserving minors (RPMs), i.e., we
    require \U0001D43B to be a minor of \U0001D43A. We show any directed graph \U0001D43A
    admits an RPM \U0001D43B of size \U0001D442(\U0001D4583), and if \U0001D43A is
    planar, then the size of \U0001D43B improves to \U0001D442(\U0001D4582log\U0001D458).
    We complement our upper bound by showing that there exists an infinite family
    of grids such that any RPM must have Ω(\U0001D4582) vertices. (2) Given a weighted
    undirected graph \U0001D43A=(\U0001D449,\U0001D438) and terminal vertices \U0001D43E
    with |\U0001D43E|=\U0001D458, an exact (vertex) cut sparsifier of \U0001D43A is
    a graph \U0001D43B with \U0001D43E⊂\U0001D449\U0001D43B that preserves the value
    of minimum cuts separating any bipartition of \U0001D43E. We show that planar
    graphs with all the \U0001D458 terminals lying on the same face admit exact cut
    sparsifiers of size \U0001D442(\U0001D4582) that are also planar. Our result extends
    to flow and distance sparsifiers. It improves the previous best-known bound of
    \U0001D442(\U0001D458222\U0001D458) for cut and flow sparsifiers by an exponential
    factor and matches an Ω(\U0001D4582) lower-bound for this class of graphs."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Gramoz
  full_name: Goranci, Gramoz
  last_name: Goranci
- first_name: Monika H
  full_name: Henzinger, Monika H
  id: 540c9bbd-f2de-11ec-812d-d04a5be85630
  last_name: Henzinger
  orcid: 0000-0002-5008-6530
- first_name: Pan
  full_name: Peng, Pan
  last_name: Peng
citation:
  ama: Goranci G, Henzinger MH, Peng P. Improved guarantees for vertex sparsification
    in planar graphs. <i>SIAM Journal on Discrete Mathematics</i>. 2020;34(1):130-162.
    doi:<a href="https://doi.org/10.1137/17m1163153">10.1137/17m1163153</a>
  apa: Goranci, G., Henzinger, M. H., &#38; Peng, P. (2020). Improved guarantees for
    vertex sparsification in planar graphs. <i>SIAM Journal on Discrete Mathematics</i>.
    Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/17m1163153">https://doi.org/10.1137/17m1163153</a>
  chicago: Goranci, Gramoz, Monika H Henzinger, and Pan Peng. “Improved Guarantees
    for Vertex Sparsification in Planar Graphs.” <i>SIAM Journal on Discrete Mathematics</i>.
    Society for Industrial &#38; Applied Mathematics, 2020. <a href="https://doi.org/10.1137/17m1163153">https://doi.org/10.1137/17m1163153</a>.
  ieee: G. Goranci, M. H. Henzinger, and P. Peng, “Improved guarantees for vertex
    sparsification in planar graphs,” <i>SIAM Journal on Discrete Mathematics</i>,
    vol. 34, no. 1. Society for Industrial &#38; Applied Mathematics, pp. 130–162,
    2020.
  ista: Goranci G, Henzinger MH, Peng P. 2020. Improved guarantees for vertex sparsification
    in planar graphs. SIAM Journal on Discrete Mathematics. 34(1), 130–162.
  mla: Goranci, Gramoz, et al. “Improved Guarantees for Vertex Sparsification in Planar
    Graphs.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 34, no. 1, Society
    for Industrial &#38; Applied Mathematics, 2020, pp. 130–62, doi:<a href="https://doi.org/10.1137/17m1163153">10.1137/17m1163153</a>.
  short: G. Goranci, M.H. Henzinger, P. Peng, SIAM Journal on Discrete Mathematics
    34 (2020) 130–162.
date_created: 2022-08-17T08:50:24Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2023-02-21T16:29:44Z
day: '01'
doi: 10.1137/17m1163153
extern: '1'
external_id:
  arxiv:
  - '1702.01136'
intvolume: '        34'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1702.01136
month: '01'
oa: 1
oa_version: Preprint
page: 130-162
publication: SIAM Journal on Discrete Mathematics
publication_identifier:
  eissn:
  - 1095-7146
  issn:
  - 0895-4801
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
related_material:
  record:
  - id: '11831'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Improved guarantees for vertex sparsification in planar graphs
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2020'
...
---
_id: '9590'
abstract:
- lang: eng
  text: We show that for any fixed dense graph G and bounded-degree tree T on the
    same number of vertices, a modest random perturbation of G will typically contain
    a copy of T . This combines the viewpoints of the well-studied problems of embedding
    trees into fixed dense graphs and into random graphs, and extends a sizeable body
    of existing research on randomly perturbed graphs. Specifically, we show that
    there is c=c(α,Δ) such that if G is an n-vertex graph with minimum degree at least
    αn, and T is an n-vertex tree with maximum degree at most Δ , then if we add cn
    uniformly random edges to G, the resulting graph will contain T asymptotically
    almost surely (as n→∞ ). Our proof uses a lemma concerning the decomposition of
    a dense graph into super-regular pairs of comparable sizes, which may be of independent
    interest.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Michael
  full_name: Krivelevich, Michael
  last_name: Krivelevich
- first_name: Matthew Alan
  full_name: Kwan, Matthew Alan
  id: 5fca0887-a1db-11eb-95d1-ca9d5e0453b3
  last_name: Kwan
  orcid: 0000-0002-4003-7567
- first_name: Benny
  full_name: Sudakov, Benny
  last_name: Sudakov
citation:
  ama: Krivelevich M, Kwan MA, Sudakov B. Bounded-degree spanning trees in randomly
    perturbed graphs. <i>SIAM Journal on Discrete Mathematics</i>. 2017;31(1):155-171.
    doi:<a href="https://doi.org/10.1137/15m1032910">10.1137/15m1032910</a>
  apa: Krivelevich, M., Kwan, M. A., &#38; Sudakov, B. (2017). Bounded-degree spanning
    trees in randomly perturbed graphs. <i>SIAM Journal on Discrete Mathematics</i>.
    Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/15m1032910">https://doi.org/10.1137/15m1032910</a>
  chicago: Krivelevich, Michael, Matthew Alan Kwan, and Benny Sudakov. “Bounded-Degree
    Spanning Trees in Randomly Perturbed Graphs.” <i>SIAM Journal on Discrete Mathematics</i>.
    Society for Industrial &#38; Applied Mathematics, 2017. <a href="https://doi.org/10.1137/15m1032910">https://doi.org/10.1137/15m1032910</a>.
  ieee: M. Krivelevich, M. A. Kwan, and B. Sudakov, “Bounded-degree spanning trees
    in randomly perturbed graphs,” <i>SIAM Journal on Discrete Mathematics</i>, vol.
    31, no. 1. Society for Industrial &#38; Applied Mathematics, pp. 155–171, 2017.
  ista: Krivelevich M, Kwan MA, Sudakov B. 2017. Bounded-degree spanning trees in
    randomly perturbed graphs. SIAM Journal on Discrete Mathematics. 31(1), 155–171.
  mla: Krivelevich, Michael, et al. “Bounded-Degree Spanning Trees in Randomly Perturbed
    Graphs.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 31, no. 1, Society
    for Industrial &#38; Applied Mathematics, 2017, pp. 155–71, doi:<a href="https://doi.org/10.1137/15m1032910">10.1137/15m1032910</a>.
  short: M. Krivelevich, M.A. Kwan, B. Sudakov, SIAM Journal on Discrete Mathematics
    31 (2017) 155–171.
date_created: 2021-06-22T12:26:25Z
date_published: 2017-01-12T00:00:00Z
date_updated: 2023-02-23T14:02:05Z
day: '12'
doi: 10.1137/15m1032910
extern: '1'
external_id:
  arxiv:
  - '1507.07960'
intvolume: '        31'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1507.07960
month: '01'
oa: 1
oa_version: Preprint
page: 155-171
publication: SIAM Journal on Discrete Mathematics
publication_identifier:
  eissn:
  - 1095-7146
  issn:
  - 0895-4801
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Bounded-degree spanning trees in randomly perturbed graphs
type: journal_article
user_id: 6785fbc1-c503-11eb-8a32-93094b40e1cf
volume: 31
year: '2017'
...