A Note on the Adversary Degree Associated Reconstruction Number of Graphs

A vertex-deleted subgraph of a graph is called a card of . A card of with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of . The degree associated reconstruction number drn ( ) of a graph is the size of the smallest collection of dacards of that uniquely determines . The adversary degree associated reconstruction number of a graph , adrn( ), is the minimum number such that every collection of dacards of that uniquely determines . In this paper, we show that adrn of wheels and complete bipartite graphs on at least 4 vertices is 2 or 3. 1. Introduction All graphs considered are simple, finite, and undirected. We will mostly follow the standard graph theoretic terminology of [1]. A vertex-deleted subgraph or card of a graph is the unlabeled graph obtained from by deleting the vertex and all edges incident with . The ordered pair is called a degree associated card or dacard of the graph where is the degree of in . The deck (dadeck) of a graph is its collection of cards (dacards). Ulam's Conjecture [2], also called Reconstruction Conjecture (RC), asserts that every graph on at least three vertices is determined uniquely (up to isomorphism) by its deck. Graphs that obey RC are called reconstructible. For a reconstructible graph , Harary and Plantholt [3] have defined the reconstruction number to be the size of the smallest subcollection of the deck of which is not contained in the deck of any other graph ; . Myrvold [4] referred to this number as ally-reconstruction number of . Myrvold [5] also studied adversary reconstruction number of which is the smallest such that no subcollection of the deck of of size is contained in the deck of any other graph ;？？ . An extension of RC to digraphs, the Digraph Reconstruction Conjecture, was disproved when Stockmeyer exhibited [6] several infinite families of counter-examples. In view of this, Ramachandran [7] studied the degree (degree triple) associated reconstruction of graphs (digraphs). For a vertex of a digraph, the ordered triple is called the degree triple of where , , and are, respectively, the number of unpaired outarcs, unpaired inarcs, and symmetric pairs of arcs incident with . A graph (digraph) is called degree associated reconstructible if it can be determined uniquely from its dadeck. For a degree associated reconstructible graph (digraph) , the degree (degree triple) associated reconstruction number, , of is the size of the smallest subcollection of the dadeck of which is not contained in the dadeck of any other graph (digraph) ;？？ . Barrus and West