Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer $h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set $V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever ${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}. Let $s(h)$ be the least integer $k$ such that the conclusion above holds and let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)= h+2^{\lfloor \log_2 h\rfloor}$. In the special case $h=2^{\ell}$ or $h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and $v(2)=6$ were obtained in a previous work.Comment: 15 page

Hypomorphy of graphs up to complementation

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 $\bar G$ 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. Ill

On the Boolean dimension of a graph and other related parameters

We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.Comment: 13 pages, 2 figure

Sur la comparaison des algebres de Boole d'intervalles

SIGLECNRS T 56312 / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

Ensembles de generateurs d'une Algebre de Boole

SIGLECNRS T Bordereau / INIST-CNRS - Institut de l'Information Scientifique et TechniqueFRFranc

{-1,2}-hypomorphy and hereditary hypomorphy coincide for posets

Let P and P' be two finite posets on the same vertex set V. The posets P and P' are hereditarily hypomorphic if for every subset X of V, the induced subposets P(X) and P'(X) are isomorphic. The posets P and P' are {-1,2}-hypomorphic if for every subset X of V, |X| in {2,|V|-1}, the subposets P(X) and P'(X) are isomorphic. P. Ille and J.X. Rampon showed that if two posets P and P', with at least 4 vertices, are {-1,2}-hypomorphic, then P and P' are isomorphic. Under the same hypothesis, we prove that P and P' are hereditarily hypomorphic. Moreover, we characterize the pairs of hereditarily hypomorphic posets

{-1,2}-hypomorphy and hereditarily hypomorphy are the same for posets

International audienceLet $P$ and $P'$ be two finite posets on the same vertex set $V$. The posets $P$ and $P'$ are {\it hereditarily hypomorphic} if for every subset $X$ of $V$, the induced subposets $P(X)$ and $P'(X)$ are isomorphic. The posets $P$ and $P'$ are $\{-1,2\}$-{\it hypomorphic} if for every subset $X$ of $V$ with $\vert X \vert \in \{2,\vert V\vert -1\}$, the subposets $P(X)$ and $P'(X)$ are isomorphic. P. Ille and J.X. Rampon \cite{Il-Ra} showed that if two posets $P$ and $P'$, with at least $4$ vertices, are $\{-1,2\}$-{hypomorphic}, then $P$ and $P'$ are isomorphic. Under the same hypothesis, we prove that $P$ and $P'$ are hereditarily hypomorphic. Moreover, we characterize the pairs of hereditarily hypomorphic posets

(-1)-Hypomorphic Graphs with the Same 3-Element Homogeneous Subsets

International audienc

{-1,2}-hypomorphy and hereditary hypomorphy coincide for posets

Let P and P' be two finite posets on the same vertex set V. The posets P and P' are hereditarily hypomorphic if for every subset X of V, the induced subposets P(X) and P'(X) are isomorphic. The posets P and P' are {-1,2}-hypomorphic if for every subset X of V, |X| in {2,|V|-1}, the subposets P(X) and P'(X) are isomorphic. P. Ille and J.X. Rampon showed that if two posets P and P', with at least 4 vertices, are {-1,2}-hypomorphic, then P and P' are isomorphic. Under the same hypothesis, we prove that P and P' are hereditarily hypomorphic. Moreover, we characterize the pairs of hereditarily hypomorphic posets