• Corpus ID: 244708976

The Implicit Graph Conjecture is False

  title={The Implicit Graph Conjecture is False},
  author={Hamed Hatami and Pooya Hatami},
An efficient implicit representation of an n-vertex graph G in a family F of graphs assigns to each vertex of G a binary code of length O(log n) so that the adjacency between every pair of vertices can be determined only as a function of their codes. This function can depend on the family but not on the individual graph. Every family of graphs admitting such a representation contains at most 2 log(n)) graphs on n vertices, and thus has at most factorial speed of growth. The Implicit Graph… 
Implicit representation of sparse hereditary families
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It is shown that, for monotone classes of graphs, there is a strict hierarchy: approximate distance threshold sketches imply small-distance sketches, which imply adjacency sketches, whereas the reverse implications are each false.
Twin-width: theory and applications
  • Biology
  • 2022
Partner Name First Current Role and Involname Position responsibilities vement LIP, MC2 BONNET Édouard CR Coordinator 36p·month LIP, MC2 CHAKRABORTY Dibyayan postdoc (É. B.) 12p·month LIP, MC2 DÉPRÉS


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It is shown that the minimum universal graph containing all bounded-degree graphs on n vertices with separators of size $n^\alpha $ has $O(n)$ edges if $\alpha \frac{1}{2}$.
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A new implicit representation of planar graphs using asymptotically 2 log n-bit labels is proposed, and all the labels can be constructed in O(n log n) time, and support constant time adjacency testing.
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Small induced-universal graphs and compact implicit graph representations
  • Stephen Alstrup, Theis Rauhe
  • Computer Science, Mathematics
    The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
  • 2002
We show that there exists a graph G with n /spl middot/ 2/sup O(log* n)/ nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies the existence of a graph
Local representations using very short labels