Ally and adversary reconstruction problems

This thesis examines the ally and adversary reconstruction problems. Both are generalizations of the famous reconstruction conjecture. The results given here provide new insight to the reconstruction conjecture, and to the development and analysis of algorithms for graph reconstruction. In this thesis, all graphs considered are simple, finite, and undirected. A vertex-deleted subgraph or card, G$-$v, of a graph G is obtained from G by deleting the vertex v and all edges incident to v. The deck of a graph G, denoted D(G), is its collection of cards. Two graphs G and H are said to have k cards in common if $\mid$D(G) $\cap$ D(H)$\mid$ = k. A graph G is reconstructible from a subset C of its deck if every graph H whose deck contains C is isomorphic to G. The problems studied can best be understood in terms of an ally and an adversary. We are given cards one at a time from a graph G. G is reconstructible from the cards given, but is not reconstructible if the last card is omitted. In the case of ally reconstruction, an ally chooses the cards so that as few cards as possible are needed to reconstruct G. For adversary reconstruction, an adversary chooses the cards so that as many cards as possible are needed. The ally-reconstruction number of a graph is the number of cards the ally selects; the adversary-reconstruction number is the number the adversary chooses. Results proved in this thesis are: (1) Almost all graphs have ally and adversary-reconstruction numbers equal to three. (2) Regular graphs can have ally and adversary-reconstruction numbers at most n/2 + 2. (3) A disconnected graph is recognizable given any n/2 + 2 cards. (4) The adversary-reconstruction problem for disconnected graphs is essentially as difficult as that for general graphs. (5) The only disconnected graphs which can have ally-reconstruction number greater than three are those whose components are all isomorphic. In this case, the ally-reconstruction number is at most c + 2 where c is the order of a component. (6) Trees on five or more vertices have ally-reconstruction number three. (7) We give a family of pairs with $n\over 2$ + $\biggl({n + 1.125\over 8}\biggr)\sp{1/2}$ +.375 common cards. No families with more common cards are known. (8) If G and H are graphs with different degree sequences on n $>$ 6 vertices, they have at most n $-$ 2 common cards.