arXiv subjects

Maurice Pouzet

Publications and source records attributed to Maurice Pouzet.

59 records · Page 4Linked to original sources

arXiv2006

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. Ille

math.CO↗

arXiv2006

From well-quasi-ordered sets to better-quasi-ordered sets

We consider conditions which force a well-quasi-ordered poset (wqo) to be better-quasi-ordered (bqo). In particular we obtain that if a poset $P$ is wqo and the set $S_ω(P)$ of strictly increasing sequences of elements of $P$ is bqo under domination, then $P$ is bqo. As a consequence, we get the same conclusion if $S_ω (P)$ is replaced by $\mathcal J^1(P)$, the collection of non-principal ideals of $P$, or by $AM(P)$, the collection of maximal antichains of $P$ ordered by domination. It then follows that an interval order which is wqo is in fact bqo.

math.CO↗

arXiv2006

Incidence structures and Stone-Priestley duality

We observe that if $R:=(I,ρ, J)$ is an incidence We observe that if $R:=(I,ρ, J)$ is an incidence structure, viewed as a matrix, then the topological closure of the set of columns is the Stone space of the Boolean algebra generated by the rows. As a consequence, we obtain that the topological closure of the collection of principal initial segments of a poset $P$ is the Stone space of the Boolean algebra $Tailalg (P)$ generated by the collection of principal final segments of $P$, the so-called {\it tail-algebra of $P$}. Similar results concerning Priestley spaces and distributive lattices are given. A generalization to incidence structures valued by abstract algebras is considered.

math.CO↗

arXiv2005

Divisibility of countable metric spaces

Prompted by a recent question of G. Hjorth as to whether a bounded Urysohn space is indivisible, that is to say has the property that any partition into finitely many pieces has one piece which contains an isometric copy of the space, we answer this question and more generally investigate partitions of countable metric spaces. We show that an indivisible metric space must be totally Cantor disconnected, which implies in particular that every Urysohn space U_V with V bounded or not but dense in some initial segment of R+, is divisible. On the other hand we also show that one can remove "large" pieces from a bounded Urysohn space with the remainder still inducing a copy of this space, providing a certain "measure" of the indivisibility. Associated with every totally Cantor disconnected space is an ultrametric space, and we go on to characterize the countable ultrametric spaces which are homogeneous and indivisible.

math.CO↗

arXiv2005

N-free extensions of posets.Note on a theorem of P.A.Grillet

Let $S\_{N}(P)$ be the poset obtained by adding a dummy vertex on each diagonal edge of the $N$'s of a finite poset $P$. We show that $S\_{N}(S\_{N}(P))$ is $N$-free. It follows that this poset is the smallest $N$-free barycentric subdivision of the diagram of $P$, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with $P\_0:=P$ and consisting at step $m$ of adding a dummy vertex on a diagonal edge of some $N$ in $P\_m$, proving that the result of this algorithm does not depend upon the particular choice of the diagonal edge choosen at each step. These results are linked to drawing of posets.

cs.DM↗