English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let  G  be any connected graph on  n  vertices,    n  ≥  2   . Let  k  be any positive integer. Suppose that a fire breaks out at some vertex of  G . Then, in each turn firefighters can protect at most  k  vertices of  G  not yet on fire; Next the fire spreads to all unprotected neighbors of burning vertices. The  k  ‐surviving rate  of G, denoted byρ  k    (  G  )    , is the expected fraction of vertices that can be saved from the fire, provided that the starting vertex is chosen uniformly at random. In this note, it is shown that for any planar graph  G  with average degree9  2   −  ε   , where    ε  ∈  (  0  ,  1  ]   , there isρ  2    (  G  )   ≥   2  9   ε   . In particular, the result implies a significant improvement of the bound for 2‐surviving rate for triangle‐free planar graphs (Esperet et al. [3][L. Esperet, 2013]) and for planar graphs without 4‐cycles (Kong et al. [12][J. Kong, 2012]). The proof is done using the separator theorem for planar graphs.

The 2‐surviving rate of planar graphs with average degree lower than 9 2