15 research outputs found

    Hypomorphy of graphs up to complementation

    Full text link
    Let VV be a set of cardinality vv (possibly infinite). Two graphs GG and GG' with vertex set VV are {\it isomorphic up to complementation} if GG' is isomorphic to GG or to the complement Gˉ\bar G of GG. Let kk be a non-negative integer, GG and GG' are {\it kk-hypomorphic up to complementation} if for every kk-element subset KK of VV, the induced subgraphs G_KG\_{\restriction K} and G_KG'\_{\restriction K} are isomorphic up to complementation. A graph GG is {\it kk-reconstructible up to complementation} if every graph GG' which is kk-hypomorphic to GG up to complementation is in fact isomorphic to GG up to complementation. We give a partial characterisation of the set S\mathcal S of pairs (n,k)(n,k) such that two graphs GG and GG' on the same set of nn vertices are equal up to complementation whenever they are kk-hypomorphic up to complementation. We prove in particular that S\mathcal S contains all pairs (n,k)(n,k) such that 4kn44\leq k\leq n-4. We also prove that 4 is the least integer kk such that every graph GG having a large number nn of vertices is kk-reconstructible up to complementation; this answers a question raised by P. Ill

    Isomorphy up to complementation

    Full text link
    Considering uniform hypergraphs, we prove that for every non-negative integer hh there exist two non-negative integers kk and tt with ktk\leq t such that two hh-uniform hypergraphs H{\mathcal H} and H{\mathcal H}' on the same set VV of vertices, with Vt| V| \geq t, are equal up to complementation whenever H{\mathcal H} and H{\mathcal H}' are kk-{hypomorphic up to complementation}. Let s(h)s(h) be the least integer kk such that the conclusion above holds and let v(h)v(h) be the least tt corresponding to k=s(h)k=s(h). We prove that s(h)=h+2log2hs(h)= h+2^{\lfloor \log_2 h\rfloor} . In the special case h=2h=2^{\ell} or h=2+1h=2^{\ell}+1, we prove that v(h)s(h)+hv(h)\leq s(h)+h. The values s(2)=4s(2)=4 and v(2)=6v(2)=6 were obtained in a previous work.Comment: 15 page

    Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

    Get PDF
    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

    Get PDF
    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]
    corecore