Size reconstructibility of graphs

Abstract

The deck of a graph G is given by the multiset of (unlabeled) subgraphs G v v V G { − : ()} ∈. The subgraphs G v − are referred to as the cards of G. Brown and Fenner recently showed that, for n 29 ≥ , the number of edges of a graph G can be computed from any deck missing 2 cards. We show that, for sufficiently large n, the number of edges can be computed from any deck missing at most n 1 20 cards.

Reconstructing the degree sequence of a sparse graph from a partial deck

“…Groenland, Guggiari and Scott [9] conjectured that the degree sequence of a graph can be reconstructed from a deck of cards with a constant number of cards missing (for n sufficiently large). It follows from Theorem 2 that the conjecture holds for graphs where the average degree is at most c k n 1 3 (for some c k depending only on k), but it is not yet known to hold for general graphs and we repeat it below.…”

“…• Groenland, Guggiari and Scott [9] proved that the number of edges can be reconstructed from any n − 1 20 √ n cards. This improves on the work of Myrvold [15] and the work of Brown and Fenner [7], who proved the bounds n − 1 and n − 2 respectively.…”

Reconstructibility of the $K_r$-count from $n-1$ cards

“…It took over 25 years for an improvement of this result. Brown and Fenner [3] showed that n − 2 cards suffice to determine the number of edges in G for sufficiently large n. Recently, Groenland, Guggari and Scott [4] showed that in fact we can miss a superconstant number of cards.…”