On a Special Case of the Generalized Neighbourhood Problem

TL;DR

This paper investigates the Generalized Neighbourhood Problem (GNP), showing that for all finite classes of graphs, the problem reduces to cases where all graphs have equal order and a fixed number of dominating vertices.

Contribution

It demonstrates a reduction of GNP and its variants to a simplified case with uniform graph order and a fixed number of dominating vertices in the class H.

Findings

GNP and its modifications are reducible to classes with equal graph orders.

The problem simplifies when all graphs in H have the same size and dominating vertices.

Reductions apply to both finite and infinite realization cases.

Abstract

For a given finite class of finite graphs H, a graph G is called a realization of H if the neighbourhood of its any vertex induces the subgraph isomorphic to a graph of H. We consider the following problem known as the Generalized Neighbourhood Problem (GNP): given a finite class of finite graphs H, does there exist a non-empty graph G that is a realization of H? In fact, there are two modifications of that problem, namely the finite (the existence of a finite realization is required) and infinite one (the realization is required to be infinite). In this paper we show that GNP and its modifications for all finite classes H of finite graphs are reduced to the same problems with an additional restriction on H. Namely, the orders of any two graphs of H are equal and every graph of H has exactly s dominating vertices.