Reconstructing graphs from their k-edge deleted subgraphs

A graph H is an edge reconstruction of the graph G if there is a bijection p from E(G) to E(H) such that for each edge e in E(G), G\e is isomorphic to EI\fi(e). (Here G\e denotes the graph obtained from G by deleting the edge e.) We call G edge reconstructible if any edge reconstruction of G is isomorphic to G. The well-known edge-reconstruction conjecture, due to Harary [4], asserts that any graph with at least four edges is edge reconstructible. If G has m edges and n vertices then it is known that it is edge-reconstructible if either 2m > (;) or 2”-’ > n! These results are due, respectively, to Lo&z [6] and Miiller [7]. In this paper we provide anologues of these results for k-edge reconstruction. A graph His a k-edge reconstruction of G if there is a bijection /3 from ($21) to (,“(_“) such that for each k-subset S of E(G), G\S is isomorphic to H\P(S). As might be expected G is k-edge reconstructible if any k-edge reconstruction of G is isomorpic to it. Our result is the following: 360 0095-8956/S? $3.00