Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of homomorphisms. As a proof of concept, we apply our method to show word-representability of the simplified graph of overlapping permutations that we introduce in this paper. For another application, we obtain results on word-representability of certain subgraphs of simplified de Bruijn graphs that were introduced recently by Petyuk and studied in the context of word-representability.

Tutte introduced planar maps in the 1960s in connection with what later became the celebrated Four-Color Theorem. A planar map is an embedding of a planar graph in the plane. Description trees, in particular, β-description trees, were introduced by Cori, Jacquard and Schaeffer in 1997, and they give a powerful tool to study planar maps. In this paper we introduce a relation on β-description trees and conjecture that this relation is a total order. Towards solving this conjecture, we provide an embedding of β(a,b)-trees into β(a−t,b+t)-trees for t≤a≤b+t, which is a far-reaching generalisation of an unpublished result of Claesson, Kitaev and Steingrímsson on embedding of β(1,0)-trees into β(0,1)-trees that gives a combinatorial proof of the fact that the number of rooted nonseparable planar maps with n+1 edges is more than the number of bicubic planar maps with 3n edges.

A new method for developing some types of distributed applications is presented. The main idea is to provide a generic tool by which a non-distributed service object can be developed into a replicated service object automatically, to provide the corresponding distributed service. A service object implements some abstract data types that can be accessed only by invoking the operations exported in its public interface. A replicated service object is implemented by a number of identical service objects running on different nodes in a network. Two challenges of our work are to exploit the semantics of service operations to maintain the replicated service object consistency, and to develop the replicated service object automatically.>