Degree Associated Edge Reconstruction Number
An edge-deleted subgraph of a graph G is called an ecard of G. An ecard of G with which the degree of the deleted edge is also given is called a degree associated ecard (or da-ecard) of G. The edeck (da-edeck) of a graph G is its collection of ecards (da-ecards). The degree associated edge reconstruction number, dern(G), of a graph G is the size of the smallest collection of ecards of G uniquely determines G. The adversary degree associated edge reconstruction number, adern(G), of a graph G is the minimum number k such that every collection of k da-ecards of G uniquely determines G. We prove that dern(G)= adern(G)=1 for any regular graph G or any bidegreed graph G with exactly one vertex of different degree, which differs by at least three. We determine dern and adern for all complete bipartite graphs except K 1,3. We also prove that dern(G)≤ 2 and adern(G)≤ 3 for any complete 3-partite graph G with n vertices in which all partite sets are equal in size as possible and a few other results.
Keywordsreconstruction number edge reconstruction number card dacard
Unable to display preview. Download preview PDF.
- 1.Harary, F.: Graph Theory. Addison Wesley, Mass. (1969)Google Scholar
- 2.Harary, F.: On the reconstruction of a graph from a collection of subgraphs. In: Fieldler, M. (ed.) Theory of Graphs and its Applications, pp. 47–52. Academic Press, New York (1964)Google Scholar
- 4.Myrvold, W.J.: The ally and adversary reconstruction problems. Ph.D. Thesis, University of Waterloo (1988)Google Scholar
- 6.Ramachandran, S.: Degree associated reconstruction number of graphs and digraphs. Mano. Int. J. Mathematical Sciences 1, 41–53 (2000)Google Scholar