@ARTICLE{TBA, 

TITLE={[FORTHCOMING] On lexicographical sum over a prime and rigid digraph, and {−1}-hypomorphy}, 
  
AUTHOR={Mouna Achour, }, 
	
JOURNAL={Advances in Pure and Applied Mathematics}, 
	
VOLUME={}, 

NUMBER={Forthcoming papers<br>}, 
	
YEAR={2026}, 

URL={https://www.openscience.fr/On-lexicographical-sum-over-a-prime-and-rigid-digraph-and-%E2%88%921-hypomorphy}, 
	
DOI={TBA}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={This work involves the reconstruction in the sense of Ulam. Let $$$G$$$ and $$$G&#039;$$$ be two digraphs with the same vertex set $$$V$$$. The digraphs $$$G$$$ and $$$G&#039;$$$ are $$$\{-1\}$$$-hypomorphic if the induced subdigraphs $$$G-x$$$ and $$$G&#039;-x$$$ are isomorphic for every vertex $$$x$$$. The digraph $$$G$$$ is $$$\{-1\}$$$-reconstructible if it is isomorphic to every digraph to which it is $$$\{-1\}$$$-hypomorphic. In a digraph $$$G$$$, a module is a vertex subset $$$M$$$ such that every vertex outside $$$M$$$ does not distinguish the vertices in $$$M$$$. For instance, $$$\emptyset$$$, $$$V(G)$$$, and the single-vertex sets are modules of $$$G$$$, called trivial modules. A module distinct from $$$V(G)$$$ is a proper module of $$$G$$$. A digraph is indecomposable if all its modules are trivial; otherwise, it is decomposable. Every indecomposable digraph with at least $$$3$$$ vertices is prime. A digraph is rigid if the unique automorphism of $$$G$$$ is the identity on $$$V(G)$$$. 
In this paper, given an $$$n$$$-vertex digraph $$$G$$$, with $$$n\geq 7$$$, that can be expressed as a lexicographical sum over a prime and rigid digraph with at most $$$n-2$$$ vertices, we describe the digraphs that are $$$\{-1\}$$$-hypomorphic to $$$G$$$, and in particular we prove that $$$G$$$ is $$$\{-1\}$$$-reconstructible.}}