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Article Dans Une Revue Discrete Mathematics, Algorithms and Applications Année : 2023

The 3-forced 2-structures

Abderrahim Boussaïri
Pierre Ille

Résumé

Given sets $S$ and $S'$, a labeled 2-structure is a function $\sigma$ from $(S\times S)\setminus\{(s,s):s\in S\}$ to $S'$. The set $S$ is called the vertex set of $\sigma$ and denoted by $V(\sigma)$. The {\em label set} of $\sigma$ is the set $L(\sigma)$ of $l\in S'$ such that $l=\sigma(v,w)$ for some $v,w\in V(\sigma)$. Given $W\subseteq V(\sigma)$, the 2-substructure $\sigma_{(\restriction (W\times W)\setminus\{(s,s):s\in W\})}$ of $\sigma$ is denoted by $\sigma[W]$. The dual $\sigma^\star$ of $\sigma$ is defined on $V(\sigma^\star)=V(\sigma)$ as follows. For distinct $v,w\in V(\sigma^\star)$, $\sigma^\star(v,w)=\sigma(w,v)$. A labeled 2-structure $\sigma$ is reversible provided that for $x,x',y,y'\in S$ such that $x\neq y$ and $x'\neq y'$, if $\sigma(x,y)=\sigma(x',y')$, then $\sigma(y,x)=\sigma(y',x')$. We only consider reversible labeled 2-structures whose vertex set is finite. Let $\sigma$ and $\tau$ be 2-structures such that $V(\sigma)=V(\tau)$. Given $2\leq k\leq|V(\sigma)|$, $\sigma$ and $\tau$ are $k$-hemimorphic if for every $W\subseteq V(\sigma)$ such that $|W|\leq k$, $\sigma[W]$ is isomorphic to $\tau[W]$ or $(\tau[W])^\star$. Furthermore, let $\sigma$ be a 2-structure. Given $2\leq k\leq|V(\sigma)|$, $\sigma$ is $k$-forced if $\sigma$ and $\sigma^\star$ are the only 2-structures $k$-hemimorphic to $\sigma$. We characterize the $3$-forced 2-structures. Lastly, we provide a large class of $4$-forced 2-structures.
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hal-04541269 , version 1 (10-04-2024)

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Abderrahim Boussaïri, Pierre Ille. The 3-forced 2-structures. Discrete Mathematics, Algorithms and Applications, 2023, ⟨10.1142/S1793830923500866⟩. ⟨hal-04541269⟩
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