A cancellation problem for rings

1973 ◽  
pp. 61-77 ◽  
Author(s):  
Paul Eakin ◽  
William Heinzer
Keyword(s):  
Cancellation Problem

scholarly journals Neighborhood Reconstruction and Cancellation of Graphs

10.37236/6676 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Richard H. Hammack ◽  
Cristina Mullican
Keyword(s):  
Direct Product ◽  
Structural Characterization ◽  
Unsolved Problem ◽  
Reconstruction Problem ◽  
Cancellation Problem ◽  
Open Vertex ◽  
Isomorphic Graphs ◽  
Product Of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

On the cancellation problem

2009 ◽  
Vol 344 (4) ◽  
pp. 769-778 ◽  
Author(s):  
Zbigniew Jelonek
Keyword(s):  
Cancellation Problem

scholarly journals Affine pseudo-planes and cancellation problem

2005 ◽  
Vol 357 (12) ◽  
pp. 4867-4883 ◽  
Author(s):  
Kayo Masuda ◽  
Masayoshi Miyanishi
Keyword(s):  
Cancellation Problem
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