When a Decomposable Tournament is $\{-1,3\}$-Reconstructible?
Résumé
Given a tournament $T$, a subset $M$ of $V(T)$ is a module of $T$ if for any $x,y\in M$ and $v\in V(T)$ such that $xv,vy\in A(T)$, we have $v\in M$. A tournament $T$ is decomposable if it admits a module $M$ such that $1<|M|<|V(T)|$. Two tournaments $T$ and $U$ are $\{-1,3\}$-hypomorphic if $T-v\simeq U-v$ for each $v\in V(T)$, and $T[X]\simeq U[X]$ for each $X\subseteq V(T)$ such that $|X|=3$. A tournament $T$ is $\{-1,3\}$-reconstructible if every tournament $\{-1,3\}$-hypomorphic to $T$ is isomorphic to $T$. We provide sufficient conditions for a decomposable tournament to be $\{-1,3\}$-reconstructible. For instance, we establish that a tournament, which admits a proper module of size $\ge 3$, is $\{-1,3\}$-reconstructible.