Hypomorphy of graphs up to complementation

Jamel Dammak (1) , Gérard Lopez (2) , Maurice Pouzet (3) , Hamza Si Kaddour (3)

1 Département de Mathematiques [Sfax]
2 IML - Institut de mathématiques de Luminy
3 ICJ - Institut Camille Jordan

Let $V$ be a set of cardinality $v$ (possibly infinite). Two graphs $G$ and $G'$ with vertex set $V$ are {\it isomorphic up to complementation} if $G'$ is isomorphic to $G$ or to the complement $\overline G$ of $G$. Let $k$ be a non-negative integer, $G$ and $G'$ are {\it $k$-hypomorphic up to complementation} if for every $k$-element subset $K$ of $V$, the induced subgraphs $G_{\restriction K}$ and $G'_{\restriction K}$ are isomorphic up to complementation. A graph $G$ is {\it $k$-reconstructible up to complementation} if every graph $G'$ which is $k$-hypomorphic to $G$ up to complementation is in fact isomorphic to $G$ up to complementation. We give a partial characterisation of the set $\mathcal S$ of pairs $(n,k)$ such that two graphs $G$ and $G'$ on the same set of $n$ vertices are equal up to complementation whenever they are $k$-hypomorphic up to complementation. We prove in particular that $\mathcal S$ contains all pairs $(n,k)$ such that $4\leq k\leq n-4$. We also prove that $4$ is the least integer $k$ such that every graph $G$ having a large number $n$ of vertices is $k$-reconstructible up to complementation; this answers a question raised by P. Ille

Mots clés: Reconstruction; Graph; Hypomorphy up to complementation; Reconstruction up to complementation; Reconstruction.

Domaines: Combinatoire [math.CO]

Jamel Dammak, Gérard Lopez, Maurice Pouzet, Hamza Si Kaddour. Hypomorphy of graphs up to complementation. 2006. ⟨hal-00016564⟩