Set-Homogeneous Graphs

Set-homogeneous directed graphs

A directed graph is set-homogeneous if, whenever U and V are isomorphic finite subdigraphs, there is an automorphism g of the digraph with Ug = V . Here, extending work of Lachlan on finite homogeneous digraphs, we classify finite set-homogeneous digraphs, where we allow some pairs of vertices to have arcs in both directions. Under the assumption that such pairs of vertices are not allowed, we ...

Set-homogeneous Graphs and Embeddings of Total Orders

For most graphs H, most H-free graphs have a linear homogeneous set

Erdős and Hajnal conjectured that for every graph H there is a constant ε = ε(H) > 0 such that every graph G that does not have H as an induced subgraph contains a clique or a stable set of order Ω(|V (G)|ε). The conjecture would be false if we set ε = 1; however, in an asymptotic setting, we obtain this strengthened form of Erdős and Hajnal’s conjecture for almost every graph H, and in particu...

Invariant measures on the set of graphs and homogeneous uncountable universal graphs

We describe the set of all invariant measures on the spaces of universal countable graphs and on the spaces of universal countable triangles-free graphs. The construction uses the description of the S∞-invariant measure on the space of infinite matrices in terms of measurable function of two variables on some special space. In its turn that space is nothing more than the universal continuous (B...

Fuzzy Forcing Set on Fuzzy Graphs

The investigation of impact of fuzzy sets on zero forcing set is the main aim of this paper. According to this, results lead us to a new concept which we introduce it as Fuzzy Zero Forcing Set (FZFS). We propose this concept and suggest a polynomial time algorithm to construct FZFS. Further more we compute the propagation time of FZFS on fuzzy graphs. This concept can be more efficient to model...