On Disconnected Graph with Large Reconstruction Number

Abstract

The reconstruction number \(rn(G)\) of graph \(G\) is the minimum number of vertex-deleted subgraphs of \(G\) required in order to identify \(G\) up to isomorphism. Myrvold and Molina have shown that if \(G\) is disconnected and not all components are isomorphic then \(rn(G) = 3\), whereas, if all components are isomorphic and have \(c\) vertices each, then \(rn(G)\) can be as large as \(c + 2\). In this paper we propose and initiate the study of the gap between \(rn(G) = 3\) and \(rn(G) = c + 2\). Myrvold showed that if \(G\) consists of \(p\) copies of \(K_c\), then\(rn(G) = c + 2\). We show that, in fact, this is the only class of disconnected graphs with this value of \(rn(G)\). We also show that if \(rn(G) \geq c + 1\) (where \(c\) is still the number of vertices in any component), then, again, \(G\) can only be copies of \(K_c\). It then follows that there exist no disconnected graphs \(G\) with \(c\) vertices in each component and \(rn(G) = c + 1\). This poses the problem of obtaining for a given \(c\), the largest value of \(t = t(c)\) such that there exists a disconnected graph with all components of order \(c\), isomorphic and not equal to \(K_c\), and is such that \(rn(G) = t\).