Bernoulli and self-destructive percolation on non-amenable graphs

In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as p approaches p_c. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.Comment: 7 page

A note on intertwines of infinite graphs

We present a construction of two infinite graphs G1, G2 and of an infinite set F of graphs such that F is an antichain with respect to the minor relation and, for every graph G in F, both G1 and G2 are subgraphs of G but no graph obtained from G by deletion or contraction of an edge has both G1 and G2 as minors. These graphs show that the extension to infinite graphs of the intertwining conjecture of Lovász, Milgram, and Ungar fails. © 1993 Academic Press, Inc

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known