15 research outputs found
Hypomorphy of graphs up to complementation
Let be a set of cardinality (possibly infinite). Two graphs and
with vertex set are {\it isomorphic up to complementation} if is
isomorphic to or to the complement of . Let be a
non-negative integer, and are {\it -hypomorphic up to
complementation} if for every -element subset of , the induced
subgraphs and are isomorphic up to
complementation. A graph is {\it -reconstructible up to complementation}
if every graph which is -hypomorphic to up to complementation is in
fact isomorphic to up to complementation. We give a partial
characterisation of the set of pairs such that two graphs
and on the same set of vertices are equal up to complementation
whenever they are -hypomorphic up to complementation. We prove in particular
that contains all pairs such that . We
also prove that 4 is the least integer such that every graph having a
large number of vertices is -reconstructible up to complementation; this
answers a question raised by P. Ill
Isomorphy up to complementation
Considering uniform hypergraphs, we prove that for every non-negative integer
there exist two non-negative integers and with such that
two -uniform hypergraphs and on the same set
of vertices, with , are equal up to complementation whenever
and are -{hypomorphic up to complementation}.
Let be the least integer such that the conclusion above holds and
let be the least corresponding to . We prove that . In the special case or
, we prove that . The values and
were obtained in a previous work.Comment: 15 page
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
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 isomorphic up to complementation if G ′ is isomorphic to G or to the complement G of G. Let k be a non-negative integer, G and G ′ are k-hypomorphic up to complementation if for every k-element subset K of V, the induced subgraphs G↾K and G ′ ↾K are isomorphic up to complementation. A graph G is 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 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 S contains all pairs (n, k) such that 4 ≤ k ≤ 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 [8]